Bayesian GLM Part1

Preparations

Load the necessary libraries

library(rstanarm) # for fitting models in STAN
library(brms) # for fitting models in STAN
library(standist) # for exploring distributions
library(coda) # for diagnostics
library(bayesplot) # for diagnostics
library(ggmcmc) # for MCMC diagnostics
library(DHARMa) # for residual diagnostics
library(rstan) # for interfacing with STAN
library(emmeans) # for marginal means etc
library(broom) # for tidying outputs
library(tidybayes) # for more tidying outputs
library(HDInterval) # for HPD intervals
library(ggeffects) # for partial plots
library(tidyverse) # for data wrangling etc
library(broom.mixed) # for summarising models
library(posterior) # for posterior draws
library(ggeffects) # for partial effects plots
library(patchwork) # for multi-panel figures
theme_set(theme_grey()) # put the default ggplot theme back
source("helperFunctions.R")

Scenario

Here is an example from Fowler, Cohen, and Jarvis (1998). An agriculturalist was interested in the effects of fertilizer load on the yield of grass. Grass seed was sown uniformly over an area and different quantities of commercial fertilizer were applied to each of ten 1 m² randomly located plots. Two months later the grass from each plot was harvested, dried and weighed. The data are in the file fertilizer.csv in the data folder.

FERTILIZER	YIELD
25	84
50	80
75	90
100	154
125	148
...	...

FERTILIZER:	Mass of fertilizer (g.m-2) - Predictor variable
YIELD:	Yield of grass (g.m-2) - Response variable

The aim of the analysis is to investigate the relationship between fertilizer concentration and grass yield.

Read in the data

We will start off by reading in the Fertilizer data. There are many functions in R that can read in a CSV file. We will use a the read_csv() function as it is part of the tidyverse ecosystem.

fert <- read_csv("../public/data/fertilizer.csv", trim_ws = TRUE)
glimpse(fert)

## Rows: 10
## Columns: 2
## $ FERTILIZER <dbl> 25, 50, 75, 100, 125, 150, 175, 200, 225, 250
## $ YIELD      <dbl> 84, 80, 90, 154, 148, 169, 206, 244, 212, 248

Exploratory data analysis

The individual responses (\(y_i\), observed yields) are each expected to have been independently drawn from normal (Gaussian) distributions (\(\mathcal{N}\)). These distributions represent all the possible yields we could have obtained at the specific (\(i^{th}\)) level of Fertilizer. Hence the \(i^{th}\) yield observation is expected to have been drawn from a normal distribution with a mean of \(\mu_i\).

Although each distribution is expected to come from populations that differ in their means, we assume that all of these distributions have the same variance (\(\sigma^2\)).

We need to supply priors for each of the parameters to be estimated (\(\beta_0\), \(\beta_1\) and \(\sigma\)). Whilst we want these priors to be sufficiently vague as to not influence the outcomes of the analysis (and thus be equivalent to the frequentist analysis), we do not want the priors to be so vague (wide) that they permit the MCMC sampler to drift off into parameter space that is both illogical as well as numerically awkward.

As a starting point, lets assign the following priors:

• \(\beta_0\): Normal prior centred at 164 with a variance of 65
• \(\beta_1\): Normal prior centred at 0 with a variance of 1
• \(\sigma\): either a half cauchy centred at 0 with a variance of 2 or an exponential with parameter of 0.016

Note, when fitting models through either rstanarm or brms, the priors assume that the predictor(s) have been centred and are to be applied on the link scale. In this case the link scale is an identity.

Model formula: \[ \begin{align} y_i &\sim{} \mathcal{N}(\mu_i, \sigma^2)\\ \mu_i &= \beta_0 + \beta_1 x_i\\ \beta_0 &\sim{} \mathcal{N}(164,65)\\ \beta_1 &\sim{} \mathcal{N}(0,1)\\ \sigma &\sim{} \mathcal{cauchy}(0,2)\\ OR\\ \sigma &\sim{} \mathcal{Exp}(0.016)\\ OR\\ \sigma &\sim{} \mathcal{gamma}(2,1)\\ \end{align} \]

Fit the model

Note, for routines that take more than a couple of seconds to perform (such as most Bayesian models), it is a good idea to cache the Rmarkdown chunks in which the routine is performed. That way, the routine will only be run the first time and any objects generated will be stored for future use. Thereafter, provided the code has not changed, the routine will not be re-run. Rather, knitr will just retrieve the cached objects and continue on.

frequentist

For the purpose of comparing the frequentist and Bayesian outcomes, it might be useful to start by fitting the frequentist simple linear model. We will also fit this model with the predictor (fertilizer concentration) centred.

summary(lm(YIELD ~ FERTILIZER, data = fert))

## 
## Call:
## lm(formula = YIELD ~ FERTILIZER, data = fert)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -22.79 -11.07  -5.00  12.00  29.79 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 51.93333   12.97904   4.001  0.00394 ** 
## FERTILIZER   0.81139    0.08367   9.697 1.07e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 19 on 8 degrees of freedom
## Multiple R-squared:  0.9216, Adjusted R-squared:  0.9118 
## F-statistic: 94.04 on 1 and 8 DF,  p-value: 1.067e-05

summary(lm(YIELD ~ scale(FERTILIZER, scale = FALSE), data = fert))

## 
## Call:
## lm(formula = YIELD ~ scale(FERTILIZER, scale = FALSE), data = fert)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -22.79 -11.07  -5.00  12.00  29.79 
## 
## Coefficients:
##                                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                      163.50000    6.00813  27.213 3.58e-09 ***
## scale(FERTILIZER, scale = FALSE)   0.81139    0.08367   9.697 1.07e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 19 on 8 degrees of freedom
## Multiple R-squared:  0.9216, Adjusted R-squared:  0.9118 
## F-statistic: 94.04 on 1 and 8 DF,  p-value: 1.067e-05

rstanarm

Using default priors

In rstanarm, the default priors are designed to be weakly informative. They are chosen to provide moderate regularisation (to help prevent over-fitting) and help stabilise the computations.

fert.rstanarm <- stan_glm(YIELD ~ FERTILIZER,
  data = fert,
  iter = 5000,
  warmup = 1000,
  chains = 3,
  thin = 5,
  refresh = 0
)

In the above:

• the formula and data arguments should be familiar as they are the same as for many models in R
• iter: indicates the number of MCMC iterations to perform per chain
• warmup: indicates how many of the initial MCMC samples to exclude. During the warmup stage, the MCMC sampler is tuning its sampling operations to allow it to determine the best way to create the chain. While it is in this discovery phase, the estimates it produces might be unrepresentative.
• chains: indicates the number of independent chains to run. The more chains, the more random starts. The point of multiple random starts is to minimise the potential for the starting point to dictate the outcome. It is possible that a chain might get stuck sampling a single feature of the posterior and not discover additional features. Additional random starting points should hopefully alleviate this. Either 3 or 4 is typical.
• thin: indicates the rate at which the MCMC samples should be thinned in order to eliminate auto-correlation between MCMC samples. When the step lengths between MCMC samples are small, the resulting parameter estimates will be auto-correlated (since samples close by in the chain will be more similar than those far apart). The rate of thinning necessary will depend on the degree of auto-correlation. For stan models, a thinning rate of 5 is a good starting point.
• refresh: indicates how often the terminal display should be updated. When running a large model, this can be useful as it provides a form of progress. However, when the routine is within an Rmarkdown document, it just results in the output including each line of progress and a lot of space is taken up unnecessarily

Having allowed rstanarm to formulate its own weakly informative priors, it is a good idea to explore what they are. Firstly, out of curiosity, it might be interesting to see what it has chosen. However, more importantly, we need to be able to document what the priors were and the rstanarm development team make it very clear that there is no guarantee that the default priors will remain the same into the future.

prior_summary(fert.rstanarm)

## Priors for model 'fert.rstanarm' 
## ------
## Intercept (after predictors centered)
##   Specified prior:
##     ~ normal(location = 164, scale = 2.5)
##   Adjusted prior:
##     ~ normal(location = 164, scale = 160)
## 
## Coefficients
##   Specified prior:
##     ~ normal(location = 0, scale = 2.5)
##   Adjusted prior:
##     ~ normal(location = 0, scale = 2.1)
## 
## Auxiliary (sigma)
##   Specified prior:
##     ~ exponential(rate = 1)
##   Adjusted prior:
##     ~ exponential(rate = 0.016)
## ------
## See help('prior_summary.stanreg') for more details

Note, consistent with good practice, rstanarm will centre all categorical predictors (whether the user centres them or not). Whilst this does not impact on estimated slope(s), it does alter the intercept (arguably for the better). Since all priors are applied on the scale of the parameters in the linear predictor (e.g. the intercept and slopes), the prior on intercept must be defined as if the predictors have been centred.

Nevertheless, rstanarm will return a posterior for intercept that is consistent with how the user had defined the linear predictor - that is, if the user does not centre the predictors, then rstanarm will return the intercept posterior on a scale consistent with as if the predictors had not been centred, even though they were automatically scaled internally.

When defining the spread of priors, rstanarm prefers to do so relative to the scaling of the data. For example, for the slope, it takes the ratio of the standard deviations of the response and the predictor and multiplies them by a factor of 2.5 (by default). Hence it is possible to define priors either in relative terms (by providing a scaling factor) or in absolute terms, by providing the desired priors along with the autoscale = FALSE argument.

The above prior summary tells us:

• for the intercept, it is using a normal prior with a mean of 164 and a standard deviation of 2.5. This mean value is based on the mean of the response variable (rounded up). The 2.5 is used for all intercepts. The scale of the prior is then adjusted proportional to the variability in the response, by multiplying 2.5 by the standard deviation of the response (and rounding up).

mean(fert$YIELD)

## [1] 163.5

sd(fert$YIELD)

## [1] 63.97439

2.5 * sd(fert$YIELD)

## [1] 159.936

• for the coefficients (in this case, just the slope), the default prior is a normal prior centred around 0 with a standard deviation of 2.5. This is then adjusted for the scale of the data by multiplying the 2.5 by the ratio of the standard deviation of the response to the standard deviation of the predictor (then rounded).

2.5 * sd(fert$YIELD) / sd(fert$FERTILIZER)

## [1] 2.113004

• the auxiliary prior represents whatever additional prior is required for the nominated model. In the case of a Gaussian model, this will be \(\sigma\), for negative binomial, it will be the reciprocal of dispersion, for gamma, it will be shape, etc . By default in rstanarm, this is a exponential with a rate of 1 which is then adjusted by division with the standard deviation of the response.

1 / sd(fert$YIELD)

## [1] 0.01563126

Assessing priors

One way to assess the priors is to have the MCMC sampler sample purely from the prior predictive distribution without conditioning on the observed data. Doing so provides a glimpse at the range of predictions possible under the priors. On the one hand, wide ranging predictions would ensure that the priors are unlikely to influence the actual predictions once they are conditioned on the data. On the other hand, if they are too wide, the sampler is being permitted to traverse into regions of parameter space that are not logically possible in the context of the actual underlying ecological context. Not only could this mean that illogical parameter estimates are possible, when the sampler is traversing regions of parameter space that are not supported by the actual data, the sampler can become unstable and have difficulty.

We can draw from the prior predictive distribution instead of conditioning on the response, by updating the model and indicating prior_PD=TRUE. After refitting the model in this way, we can plot the predictions to gain insights into the range of predictions supported by the priors alone.

fert.rstanarm1 <- update(fert.rstanarm, prior_PD = TRUE)

ggpredict(fert.rstanarm1) %>% plot(add.data = TRUE)

## $FERTILIZER

Conclusions:

• we see that the range of predictions is fairly wide and the slope could range from strongly negative to strongly positive.
• at the same time, massive yields are not supported, so there is some constraints on the parameters space
• nevertheless, negative yields are considered possible, even though these are clearly not logical.

Defining priors

The following link provides some guidance about defining priors. [https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations]

When defining our own priors, we typically do not want them to be scaled.

If we wanted to define our own priors that were less vague, yet still not likely to bias the outcomes, we could try the following priors (mainly plucked out of thin air):

• \(\beta_0\): normal centred at 164 with a standard deviation of 65
• \(\beta_1\): normal centred at 0 with a standard deviation of 1
• \(\sigma\): half-cauchy centred at 0 with a standard deviation of 2

We can visualise each of the above as:

standist::visualize("normal(164, 65)", xlim = c(0, 300))

standist::visualize("normal(0, 1)", xlim = c(-10, 10))

standist::visualize("cauchy(0, 2)",
  "gamma(2,1)",
  "exponential(0.016)",
  xlim = c(-10, 50)
)

I will also overlay the raw data for comparison.

fert.rstanarm2 <- stan_glm(YIELD ~ FERTILIZER,
  data = fert,
  prior_intercept = normal(164, 65, autoscale = FALSE),
  prior = normal(0, 1, autoscale = FALSE),
  prior_aux = cauchy(0, 2, autoscale = FALSE),
  prior_PD = TRUE,
  iter = 5000, warmup = 1000,
  chains = 3, thin = 5, refresh = 0
)

ggpredict(fert.rstanarm2) %>%
  plot(add.data = TRUE)

## $FERTILIZER

Now lets refit, conditioning on the data.

fert.rstanarm3 <- update(fert.rstanarm2, prior_PD = FALSE)

Plotting prior and posterior

posterior_vs_prior(fert.rstanarm3,
  color_by = "vs", group_by = TRUE,
  facet_args = list(scales = "free_y")
)

Conclusions:

• in each case, the prior is substantially wider than the posterior, suggesting that the posterior is not biased towards the prior.

ggpredict(fert.rstanarm3) %>% plot(add.data = TRUE)

## $FERTILIZER

brms

Using default priors

In brms, the default priors are designed to be weakly informative. They are chosen to provide moderate regularisation (to help prevent over-fitting) and help stabilise the computations.

Unlike rstanarm, brms models must be compiled before they start sampling. For most models, the compilation of the stan code takes around 45 seconds.

fert.brm <- brm(bf(YIELD ~ FERTILIZER),
  data = fert,
  iter = 5000,
  warmup = 1000,
  chains = 3,
  thin = 5,
  refresh = 0
)

In the above:

• the formula and data arguments should be familiar as they are the same as for many models in R. That said, the formula specification of brms can define models that are not possible by most other routines. To facilitate this enhanced functionality, we usually define a brms formula within its own bf() function along with the family (in this case, it is Gaussian, which is the default and therefore can be omitted.)
• iter: indicates the number of MCMC iterations to perform per chain
• warmup: indicates how many of the initial MCMC samples to exclude. During the warmup stage, the MCMC sampler is tuning its sampling operations to allow it to determine the best way to create the chain. While it is in this discovery phase, the estimates it produces might be unrepresentative.
• chains: indicates the number of independent chains to run. The more chains, the more random starts. The point of multiple random starts is to minimise the potential for the starting point to dictate the outcome. It is possible that a chain might get stuck sampling a single feature of the posterior and not discover additional features. Additional random starting points should hopefully alleviate this. Either 3 or 4 is typical.
• thin: indicates the rate at which the MCMC samples should be thinned in order to eliminate auto-correlation between MCMC samples. When the step lengths between MCMC samples are small, the resulting parameter estimates will be auto-correlated (since samples close by in the chain will be more similar than those far apart). The rate of thinning necessary will depend on the degree of auto-correlation. For stan models, a thinning rate of 5 is a good starting point.
• refresh: indicates how often the terminal display should be updated. When running a large model, this can be useful as it provides a form of progress. However, when the routine is within an Rmarkdown document, it just results in the output including each line of progress and a lot of space is taken up unnecessarily

The returned object (a list) contains a range of objects. The output from rstan is contained in the fit list item.

fert.brm %>% names()

##  [1] "formula"   "data"      "prior"     "data2"     "stanvars"  "model"    
##  [7] "ranef"     "save_pars" "algorithm" "backend"   "threads"   "opencl"   
## [13] "fit"       "criteria"  "file"      "version"   "family"    "autocor"  
## [19] "cov_ranef" "stan_funs" "data.name"

Having allowed brms to formulate its own weakly informative priors, it is a good idea to explore what they are. Firstly, out of curiosity, it might be interesting to see what it has chosen. However, more importantly, we need to be able to document what the priors were and the brms development team make it very clear that there is no guarantee that the default priors will remain the same into the future.

prior_summary(fert.brm)

##                      prior     class       coef group resp dpar nlpar bound       source
##                     (flat)         b                                             default
##                     (flat)         b FERTILIZER                             (vectorized)
##  student_t(3, 161.5, 90.4) Intercept                                             default
##      student_t(3, 0, 90.4)     sigma                                             default

This tells us:

• for the intercept, it is using a student t (flatter normal) prior with a mean of 161.5 and a standard deviation of 90.4. These mean and standard deviation values are based on the median and median absolute deviation of the response variable (rounded).

median(fert$YIELD)

## [1] 161.5

mad(fert$YIELD)

## [1] 90.4386

mad is the median absolute deviation. mad is to var as median is to mean.

standist::visualize("student_t(3,161.5,90.4)", xlim = c(-100, 1000))

• for the beta coefficients (in this case, just the slope), the default prior is a improper flat prior. A flat prior essentially means that any value between negative infinity and positive infinity are equally likely. Whilst this might seem reckless, in practice, it seems to work reasonably well for non-intercept beta parameters.

• the prior on sigma is also a student t (although only the positive half is used in the stan code), with mean of 0 and standard deviation of 90.4.

standist::visualize("student_t(3,0,90.4)", xlim = c(-10, 500))

Assessing priors

One way to assess the priors is to have the MCMC sampler sample purely from the prior predictive distribution without conditioning on the observed data. Doing so provides a glimpse at the range of predictions possible under the priors. On the one hand, wide ranging predictions would ensure that the priors are unlikely to influence the actual predictions once they are conditioned on the data. On the other hand, if they are too wide, the sampler is being permitted to traverse into regions of parameter space that are not logically possible in the context of the actual underlying ecological context. Not only could this mean that illogical parameter estimates are possible, when the sampler is traversing regions of parameter space that are not supported by the actual data, the sampler can become unstable and have difficulty.

In brms, we can inform the sampler to draw from the prior predictive distribution instead of conditioning on the response, by running the model with the sample_prior = 'only' argument. Unfortunately, this cannot be applied when there are flat priors (since the posteriors will necessarily extend to negative and positive infinity). Therefore, in order to use this useful routine, we need to make sure that we have defined a proper prior for all parameters.

Lets try a Gaussian (normal) distribution for the slope. We will start with what is likely to be a very wide distribution (wide with respect to what we know about the likely slope).

standist::visualize("normal(0, 10)", xlim = c(-50, 50))

standist::visualize("normal(0, 0.1)", xlim = c(-1, 1))

fert.brm1 <- brm(bf(YIELD ~ FERTILIZER),
  data = fert,
  prior = prior(normal(0, 10), class = "b"),
  sample_prior = "only",
  iter = 5000,
  warmup = 1000,
  chains = 3,
  thin = 5,
  refresh = 0
)

fert.brm1 %>%
  ggpredict() %>%
  plot(add.data = TRUE)

## $FERTILIZER

fert.brm1 %>%
  ggemmeans(~FERTILIZER) %>%
  plot(add.data = TRUE)

fert.brm1 %>%
  conditional_effects() %>%
  plot(points = TRUE)

Conclusions:

• we see that the range of predictions is fairly wide and the slope could range from strongly negative to strongly positive.
• at the same time, massive yields are not supported, so there is some constraints on the parameters space
• nevertheless, negative yields are considered possible, even though these are clearly not logical.

Defining priors

The following link provides some guidance about defining priors. [https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations]

When defining our own priors, we typically do not want them to be scaled.

If we wanted to define our own priors that were less vague, yet still not likely to bias the outcomes, we could try the following priors (which I have mainly plucked out of thin air):

• \(\beta_0\): normal centred at 164 with a standard deviation of 65
• \(\beta_1\): normal centred at 0 with a standard deviation of 1
• \(\sigma\): half-cauchy centred at 0 with a standard deviation of 2
• \(\sigma\): gamma with shape parameters of 2 and 1

standist::visualize("normal(164, 65)", xlim = c(-100, 300))

standist::visualize("normal(0, 1)", xlim = c(-10, 10))

standist::visualize("cauchy(0, 2)", xlim = c(-10, 10))

standist::visualize("gamma(2, 1)", xlim = c(0, 10))

standist::visualize(
  "student_t(3, 0, 90.4)",
  "cauchy(0, 2)",
  "gamma(2, 1)",
  "gamma(2, 0.5)",
  "gamma(2, 0.2)",
  xlim = c(0, 20)
)

I will also overlay the raw data for comparison.

priors <- prior(normal(164, 65), class = "Intercept") +
  prior(normal(0, 1), class = "b") +
  prior(gamma(2, 1), class = "sigma")

fert.brm2 <- brm(bf(YIELD ~ FERTILIZER),
  data = fert,
  prior = priors,
  sample_prior = "only",
  iter = 5000,
  warmup = 1000,
  chains = 3,
  thin = 5,
  refresh = 0
)

fert.brm2 %>%
  ggpredict() %>%
  plot(add.data = TRUE)

## $FERTILIZER

# OR
fert.brm2 %>%
  ggemmeans(~FERTILIZER) %>%
  plot(add.data = TRUE)

# OR
fert.brm2 %>%
  conditional_effects() %>%
  plot(points = TRUE)

fert.brm3 <- update(fert.brm2, sample_prior = "yes", refresh = 0)
# OR
fert.brm3 <- brm(bf(YIELD ~ FERTILIZER),
  data = fert,
  prior = priors,
  sample_prior = "yes",
  iter = 5000,
  warmup = 1000,
  chains = 3,
  thin = 5,
  refresh = 0
)

Plotting prior and posterior

When we have indicated that the posterior should be informed by both the prior and the posterior, both prior (governed by priors alone) and posterior (governed by both priors and data/likelihood) draws are returned. These can be compared by exploring the probabilities associated with specific hypotheses - the most obvious of which is that of no effect (that the parameter = 0).

When doing so, ideally the posterior should be distinct from the prior. If this is not the case, it might suggest that the posteriors are being too strongly driven by the priors.

fert.brm3 %>% get_variables()

##  [1] "b_Intercept"     "b_FERTILIZER"    "sigma"           "prior_Intercept"
##  [5] "prior_b"         "prior_sigma"     "lp__"            "accept_stat__"  
##  [9] "stepsize__"      "treedepth__"     "n_leapfrog__"    "divergent__"    
## [13] "energy__"

## fert.brm3 %>% hypothesis('Intercept=0', class='b') %>% plot
fert.brm3 %>%
  hypothesis("FERTILIZER = 0") %>%
  plot()

## fert.brm3 %>% hypothesis('scaleFERTILIZERscaleEQFALSE=0') %>% plot
fert.brm3 %>%
  hypothesis("sigma = 0", class = "") %>%
  plot()

Unfortunately, it is not possible to do this comparison sensibly for the intercept. The reason for this is that the prior for intercept was applied to an intercept that is associated with centred continuous predictors (predictors are centred behind the scenes). Since we did not centre the predictor, the intercept returned is as if uncentred. Hence, the prior and posterior are on different scales.

fert.brm3 %>% get_variables()

##  [1] "b_Intercept"     "b_FERTILIZER"    "sigma"           "prior_Intercept"
##  [5] "prior_b"         "prior_sigma"     "lp__"            "accept_stat__"  
##  [9] "stepsize__"      "treedepth__"     "n_leapfrog__"    "divergent__"    
## [13] "energy__"

fert.brm3 %>% SUYR_prior_and_posterior()

Exploring the stan code

fert.brm3 %>% standata()

## $N
## [1] 10
## 
## $Y
##  [1]  84  80  90 154 148 169 206 244 212 248
## 
## $K
## [1] 2
## 
## $X
##    Intercept FERTILIZER
## 1          1         25
## 2          1         50
## 3          1         75
## 4          1        100
## 5          1        125
## 6          1        150
## 7          1        175
## 8          1        200
## 9          1        225
## 10         1        250
## attr(,"assign")
## [1] 0 1
## 
## $prior_only
## [1] 0
## 
## attr(,"class")
## [1] "standata" "list"

fert.brm3 %>% stancode()

## // generated with brms 2.16.3
## functions {
## }
## data {
##   int<lower=1> N;  // total number of observations
##   vector[N] Y;  // response variable
##   int<lower=1> K;  // number of population-level effects
##   matrix[N, K] X;  // population-level design matrix
##   int prior_only;  // should the likelihood be ignored?
## }
## transformed data {
##   int Kc = K - 1;
##   matrix[N, Kc] Xc;  // centered version of X without an intercept
##   vector[Kc] means_X;  // column means of X before centering
##   for (i in 2:K) {
##     means_X[i - 1] = mean(X[, i]);
##     Xc[, i - 1] = X[, i] - means_X[i - 1];
##   }
## }
## parameters {
##   vector[Kc] b;  // population-level effects
##   real Intercept;  // temporary intercept for centered predictors
##   real<lower=0> sigma;  // dispersion parameter
## }
## transformed parameters {
## }
## model {
##   // likelihood including constants
##   if (!prior_only) {
##     target += normal_id_glm_lpdf(Y | Xc, Intercept, b, sigma);
##   }
##   // priors including constants
##   target += normal_lpdf(b | 0, 1);
##   target += normal_lpdf(Intercept | 164, 65);
##   target += gamma_lpdf(sigma | 2, 1);
## }
## generated quantities {
##   // actual population-level intercept
##   real b_Intercept = Intercept - dot_product(means_X, b);
##   // additionally sample draws from priors
##   real prior_b = normal_rng(0,1);
##   real prior_Intercept = normal_rng(164,65);
##   real prior_sigma = gamma_rng(2,1);
##   // use rejection sampling for truncated priors
##   while (prior_sigma < 0) {
##     prior_sigma = gamma_rng(2,1);
##   }
## }

INLA

library(INLA)

Using default priors

In INLA, the default priors are designed to be diffuse or weak. They are chosen to provide moderate regularisation (to help prevent over-fitting) and help stabilise the computations.

fert.inla <- inla(YIELD ~ FERTILIZER,
  data = fert,
  family = "gaussian",
  control.compute = list(config = TRUE, dic = TRUE, waic = TRUE, cpo = TRUE)
)

In the above:

• the formula, data and family arguments should be familiar as they are the same as for other models in R.

• control.compute: allows us to indicate what additional actions should be performed during the model fitting. In this case, we have indicated:

  • dic: Deviance information criterion
  • waic: Wantanabe information creterion
  • cpo: out-of-sample estimates (measures of fit)
  • config: return the full configuration - to allow drawing from the posterior.

fert.inla %>% names()

##  [1] "names.fixed"                 "summary.fixed"              
##  [3] "marginals.fixed"             "summary.lincomb"            
##  [5] "marginals.lincomb"           "size.lincomb"               
##  [7] "summary.lincomb.derived"     "marginals.lincomb.derived"  
##  [9] "size.lincomb.derived"        "mlik"                       
## [11] "cpo"                         "po"                         
## [13] "waic"                        "model.random"               
## [15] "summary.random"              "marginals.random"           
## [17] "size.random"                 "summary.linear.predictor"   
## [19] "marginals.linear.predictor"  "summary.fitted.values"      
## [21] "marginals.fitted.values"     "size.linear.predictor"      
## [23] "summary.hyperpar"            "marginals.hyperpar"         
## [25] "internal.summary.hyperpar"   "internal.marginals.hyperpar"
## [27] "offset.linear.predictor"     "model.spde2.blc"            
## [29] "summary.spde2.blc"           "marginals.spde2.blc"        
## [31] "size.spde2.blc"              "model.spde3.blc"            
## [33] "summary.spde3.blc"           "marginals.spde3.blc"        
## [35] "size.spde3.blc"              "logfile"                    
## [37] "misc"                        "dic"                        
## [39] "mode"                        "neffp"                      
## [41] "joint.hyper"                 "nhyper"                     
## [43] "version"                     "Q"                          
## [45] "graph"                       "ok"                         
## [47] "cpu.used"                    "all.hyper"                  
## [49] ".args"                       "call"                       
## [51] "model.matrix"

Having allowed INLA to formulate its own “minimally informative” priors, it is a good idea to explore what they are. Firstly, out of curiosity, it might be interesting to see what it has chosen. However, more importantly, we need to be able to document what the priors were and the INLA development team make it very clear that there is no guarantee that the default priors will remain the same into the future.

In calcutating the posterior mode of hyperparameters, it is efficient to maximising the sum of the (log)-likelihood and the (log)-prior, hence, priors are defined on a log-scale. The canonical prior for variance is the gamma prior, hence in INLA, this is a loggamma.

They are also defined according to their mean and precision (inverse-scale, rather than variance). Precision is \(1/\sigma\).

To explore the default priors used by INLA, we can issue the following on the fitted model:

inla.priors.used(fert.inla)

## section=[family]
##  tag=[INLA.Data1] component=[gaussian]
##      theta1:
##          parameter=[log precision]
##          prior=[loggamma]
##          param=[1e+00, 5e-05]
## section=[fixed]
##  tag=[(Intercept)] component=[(Intercept)]
##      beta:
##          parameter=[(Intercept)]
##          prior=[normal]
##          param=[0, 0]
##  tag=[FERTILIZER] component=[FERTILIZER]
##      beta:
##          parameter=[FERTILIZER]
##          prior=[normal]
##          param=[0.000, 0.001]

The above indicates:

• the prior on the overal family (log) precision (\(1/sigma\)) is a log-gamma with shape of 1 and scale of 0.00005. Since this is a log-gamma on log-precision, it is equivalent to a gamma (shape = \(1\), scale = \(10^-5\)) on precision. Such a distribution is very wide and has a mean of (\(shape/scale =\) 2^{4}) and variance of (\(shape/scale^2 =\) 4^{8}).

standist::visualize("gamma(1, 0.00005)", xlim=c(-100,100000))

• the prior on the intercept is a Gaussian (normal) distribution with mean of 0 and precision of 0 - this is effectively a uniform distribution from \(-\infty\) to \(\infty\).
• the prior on the slope is a Gaussian (normal) distribution with a mean of 0 and a precision of 0.001 (which equates to a standard deviation of \((1/0.001)^{0.5}\)=31.6227766. This is also a very wide distribution (consider what a slope large in magnitude of 100 means in the context of the rate of grass yield increase per unit increase in fertiliser concentration).

standist::visualize("normal(0, 31)", xlim=c(-100,100))

Defining own priors

Family variance

In order to drive the specification of weekly informative priors for variance components (a prior that allows the data to dictate the parameter estimates whilst constraining the range of the marginal distribution within sensible bounds), we can consider the range over which the marginal distribution of variance components would be sensible. For the range [-R,R], the equivalent gamma parameters:

\[ \begin{align} a &= \frac{d}{2}\\ b &= \frac{R^2d}{2*(t^d_{1-(1-q)/2})^2} \end{align} \]

where \(d\) is the degrees of freedom (for Cauchy marginal, \(d=1\)) and \(t^d_{1-(1-q)/2}\) is the \(q_{th}\) quantile of a Student t with \(d\) degrees of freedom. Hence if we considered variance likely to be in the range of [0,10], we could define a loggamma prior of:

\[ \begin{align} a &= \frac{1}{2} = 0.5\\ b &= \frac{10^2}{2*161.447} = 0.31\\ \tau &\sim{} log\Gamma(0.5,0.31) \end{align} \]

standist::visualize("gamma(0.5, 0.31)", xlim=c(-5,20))

Intercept

The default prior on the intercept is a Gaussian with mean of 0 and precision of 0 (and thus effectively a flat uniform). Alternatively, we could define priors on the intercept that are more realistic. For example, we know that the median grass yield is 80. Hence, the intercept is likely to be relatively close to this value. Note, arguably it would be more sensible to center the predictor variable (fertiliser concentration) so that the intercept would represent the yeild at the average fertiliser concentration. Furthermore, this would also be more computationally expedient. In fact, the only reason we are not doing so is so that we can directly compare the parameter estimates to the frequentist linear model.

min(fert$YIELD)

## [1] 80

median(fert$YIELD)

## [1] 161.5

mad(fert$YIELD)

## [1] 90.4386

We will multiply the variability (90.4386) by 3 so as to estimate an envelope that so as to fully capture the full range of expected values (271.3158.

standist::visualize("normal(80, 271.32)", xlim=c(-700,1000))

We could use these values as the basis for weekly informative priors on the intercept. Note, as INLA priors are expressed in terms of precision rather than variance, an equvilent prior would be \(\sim{}~\mathcal{N}(80, 0.00001)\) (e.g. \(1/(MAD\times 3)^2\)).

Fixed effects

The priors for the fixed effects (slope) is a Gaussian (normal) distributions with mean of 0 and precision (0.001). This implies that the prior for slope has a standard deviation of approximately 31 (since \(\sigma = \frac{1}{\sqrt{\tau}}\)). As a general rule, three standard deviations envelopes most of a distribution, and thus this prior defines a distribution whose density is almost entirely within the range [-93,93]. Arguably, this is very wide for the slope given the range of the predictor variable.

In order to generate realistic informative Gaussian priors (for the purpose of constraining the posterior to a logical range) for fixed parameters, the following formulae are useful:

\[ \begin{align} \mu &= \frac{z_2\theta_1 - z_1\theta_2}{z_2-z_1}\\ \sigma &= \frac{\theta_2 - \theta_1}{z_2-z_1} \end{align} \]

where \(\theta_1\) and \(\theta_2\) are the quantiles on the response scale and \(z_1\) and \(z_2\) are the corresponding quantiles on the standard normal scale. Hence, if we considered that the slope is likely to be in the range of [-10,10], we could specify a Normal prior with mean of \(\mu=\frac{(qnorm(0.5,0,1)*0) - (qnorm(0.975,0,1)*10)}{10-0} = 0\) and a standard deviation of \(\sigma^2=\frac{10 - 0}{qnorm(0.975,0,1)-qnorm(0.5,0,1)} = 5.102\). In INLA (which defines priors in terms of precision rather than standard deviation), the associated prior would be \(\beta \sim{} \mathcal{N}(0, 0.0384)\).

standist::visualize("normal(0, 5.102)", xlim=c(-20,20))

In order to define each of the above priors, we could modify the inla call:

fert.inla1 <- inla(YIELD ~ FERTILIZER,
  data = fert,
  family = "gaussian",
  control.fixed = list(
    mean.intercept = 80,
    prec.intercept = 0.00001,
    mean = 0,
    prec = 0.0384
  ),
  control.family = list(hyper = list(prec = list(
    prior = "loggamma",
    param = c(0.5, 0.31)
  ))),
  control.compute = list(config = TRUE, dic = TRUE, waic = TRUE, cpo = TRUE)
)

Lets briefly compare the parameter estimates. In the following, Model.1 is the model that employed the default priors and Model.2 is the model that employed the user defined priors.

## Fixed effects
rbind(Model.1 = fert.inla$summary.fixed, Model.2 = fert.inla1$summary.fixed)

## Family hyperpriors (variance)
rbind(Model.1 = fert.inla$summary.hyperpar, Model.2 = fert.inla1$summary.hyperpar)

It is clear that the two models yeild very similar parameter estimates.

Plotting prior vs posterior

Note, the following will not appear in this tutorial as the INLA plotting function has been really awkwardly written and is not compatible with Rmd.

plot(fert.inla1,
  plot.fixed.effects = TRUE,
  plot.lincomb = FALSE,
  plot.random.effects = FALSE,
  plot.hyperparameters = TRUE,
  plot.predictor = FALSE,
  plot.q = FALSE,
  plot.cpo = FALSE,
  plot.prior = TRUE,
  single = FALSE
)

library(INLAutils)
plot_fixed_marginals(fert.inla1, priors = TRUE)

library(INLAutils)
plot_hyper_marginals(fert.inla1, CI = TRUE)

Or the effects as a single figure..

library(INLAutils)
autoplot(fert.inla1)

marg.fix <- fert.inla1$marginals.fixed
(nm <- names(marg.fix))

## [1] "(Intercept)" "FERTILIZER"

p <- vector("list", length(nm))
for (wch in 1:length(nm)) {
  xy <- INLA:::inla.get.prior.xy(
    section = "fixed", hyperid = nm[1],
    all.hyper = fert.inla1$all.hyper,
    range = 3 * range(marg.fix[[wch]][, "x"])
  )
  p[[wch]] <- ggplot() +
    geom_density(
      data = as.data.frame(marg.fix[[wch]]),
      aes(x = x, y = y, fill = "Posterior"), alpha = 0.2,
      stat = "Identity"
    ) +
    geom_density(
      data = as.data.frame(xy), aes(x = x, y = y, fill = "Prior"), alpha = 0.2,
      stat = "Identity"
    )
}
patchwork::wrap_plots(p)

MCMC sampling diagnostics

MCMC sampling behaviour

Since the purpose of the MCMC sampling is to estimate the posterior of an unknown joint likelihood, it is important that we explore a range of diagnostics designed to help identify when the resulting likelihood might not be accurate.

• traceplots - plots of the individual draws in sequence. Traces that resemble noise suggest that all likelihood features are likely to have be traversed. Obvious steps or blocks of noise are likely to represent distinct features and could imply that there are yet other features that have not yet been traversed - necessitating additional iterations. Furthermore, each chain should be indistinguishable from the others
• autocorrelation function - plots of the degree of correlation between pairs of draws for a range of lags (distance along the chains). High levels of correlation (after a lag of 0, which is correlating each draw with itself) suggests a lack of independence between the draws and that therefore, summaries such as mean and median will be biased estimates. Ideally, all non-zero lag correlations should be less than 0.2.
• convergence diagnostics - there are a range of diagnostics aimed at exploring whether the multiple chains are likely to have converged upon similar posteriors
  • R hat - this metric compares between and within chain model parameter estimates, with the expectation that if the chains have converged, the between and within rank normalised estimates should be very similar (and Rhat should be close to 1). The more one chains deviates from the others, the higher the Rhat value. Values less than 1.05 are considered evidence of convergence.
  • Bulk ESS - this is a measure of the effective sample size from the whole (bulk) of the posterior and is a good measure of the sampling efficiency of draws across the entire posterior
  • Tail ESS - this is a measure of the effective sample size from the 5% and 95% quantiles (tails) of the posterior and is a good measure of the sampling efficiency of draws from the tail (areas of the posterior with least support and where samplers can get stuck).

available_mcmc()

Package	Description	function	rstanarm	brms
bayesplot	Traceplot	mcmc_trace	plot(mod, plotfun='trace')	mcmc_plot(mod, type='trace')
	Density plot	mcmc_dens	plot(mod, plotfun='dens')	mcmc_plot(mod, type='dens')
	Density & Trace	mcmc_combo	plot(mod, plotfun='combo')	mcmc_plot(mod, type='combo')
	ACF	mcmc_acf_bar	plot(mod, plotfun='acf_bar')	mcmc_plot(mod, type='acf_bar')
	Rhat hist	mcmc_rhat_hist	plot(mod, plotfun='rhat_hist')	mcmc_plot(mod, type='rhat_hist')
	No. Effective	mcmc_neff_hist	plot(mod, plotfun='neff_hist')	mcmc_plot(mod, type='neff_hist')
rstan	Traceplot	stan_trace	stan_trace(mod)	stan_trace(mod)
	ACF	stan_ac	stan_ac(mod)	stan_ac(mod)
	Rhat	stan_rhat	stan_rhat(mod)	stan_rhat(mod)
	No. Effective	stan_ess	stan_ess(mod)	stan_ess(mod)
	Density plot	stan_dens	stan_dens(mod)	stan_dens(mod)
ggmcmc	Traceplot	ggs_traceplot	ggs_traceplot(ggs(mod))	ggs_traceplot(ggs(mod))
	ACF	ggs_autocorrelation	ggs_autocorrelation(ggs(mod))	ggs_autocorrelation(ggs(mod))
	Rhat	ggs_Rhat	ggs_Rhat(ggs(mod))	ggs_Rhat(ggs(mod))
	No. Effective	ggs_effective	ggs_effective(ggs(mod))	ggs_effective(ggs(mod))
	Cross correlation	ggs_crosscorrelation	ggs_crosscorrelation(ggs(mod))	ggs_crosscorrelation(ggs(mod))
	Scale reduction	ggs_grb	ggs_grb(ggs(mod))	ggs_grb(ggs(mod))

In addition to the regular model diagnostics checking, for Bayesian analyses, it is also necessary to explore the MCMC sampling diagnostics to be sure that the chains are well mixed and have converged on a stable posterior.

There are a wide variety of tests that range from the big picture, overall chain characteristics to the very specific detailed tests that allow the experienced modeller to drill down to the very fine details of the chain behaviour. Furthermore, there are a multitude of packages and approaches for exploring these diagnostics.

rstanarm

bayesplot

The bayesplot package offers a range of MCMC diagnostics as well as Posterior Probability Checks (PPC), all of which have a convenient plot() interface. Lets start with the MCMC diagnostics.

available_mcmc()

## bayesplot MCMC module:
##   mcmc_acf
##   mcmc_acf_bar
##   mcmc_areas
##   mcmc_areas_data
##   mcmc_areas_ridges
##   mcmc_areas_ridges_data
##   mcmc_combo
##   mcmc_dens
##   mcmc_dens_chains
##   mcmc_dens_chains_data
##   mcmc_dens_overlay
##   mcmc_hex
##   mcmc_hist
##   mcmc_hist_by_chain
##   mcmc_intervals
##   mcmc_intervals_data
##   mcmc_neff
##   mcmc_neff_data
##   mcmc_neff_hist
##   mcmc_nuts_acceptance
##   mcmc_nuts_divergence
##   mcmc_nuts_energy
##   mcmc_nuts_stepsize
##   mcmc_nuts_treedepth
##   mcmc_pairs
##   mcmc_parcoord
##   mcmc_parcoord_data
##   mcmc_rank_hist
##   mcmc_rank_overlay
##   mcmc_recover_hist
##   mcmc_recover_intervals
##   mcmc_recover_scatter
##   mcmc_rhat
##   mcmc_rhat_data
##   mcmc_rhat_hist
##   mcmc_scatter
##   mcmc_trace
##   mcmc_trace_data
##   mcmc_trace_highlight
##   mcmc_violin

Of these, we will focus on:

• mcmc_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different shade of blue, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).

plot(fert.rstanarm3, plotfun = "mcmc_trace")

The chains appear well mixed and very similar

• acf (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain

plot(fert.rstanarm3, "acf_bar")

There is no evidence of autocorrelation in the MCMC samples

• Rhat: Rhat is a measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.

plot(fert.rstanarm3, "rhat_hist")

All Rhat values are below 1.05, suggesting the chains have converged.

• neff (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

  If the ratios are low, tightening the priors may help.

plot(fert.rstanarm3, "neff_hist")

Ratios all very high.

More diagnostics

plot(fert.rstanarm3, "combo")

plot(fert.rstanarm3, "violin")

stan plots

The rstan package offers a range of MCMC diagnostics. Lets start with the MCMC diagnostics.

Of these, we will focus on:

• stan_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).

stan_trace(fert.rstanarm3)

The chains appear well mixed and very similar

• stan_acf (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain

stan_ac(fert.rstanarm3)

There is no evidence of autocorrelation in the MCMC samples

• stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.

stan_rhat(fert.rstanarm3)

All Rhat values are below 1.05, suggesting the chains have converged.

• stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

  If the ratios are low, tightening the priors may help.

stan_ess(fert.rstanarm3)

Ratios all very high.

stan_dens(fert.rstanarm3, separate_chains = TRUE)

ggmcmc

The ggmean package also has a set of MCMC diagnostic functions. Lets start with the MCMC diagnostics.

Of these, we will focus on:

• ggs_traceplot: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).

fert.ggs <- ggs(fert.rstanarm3)
ggs_traceplot(fert.ggs)

The chains appear well mixed and very similar

• gss_autocorrelation (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain

ggs_autocorrelation(fert.ggs)

There is no evidence of autocorrelation in the MCMC samples

• stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.

ggs_Rhat(fert.ggs)

All Rhat values are below 1.05, suggesting the chains have converged.

• stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

  If the ratios are low, tightening the priors may help.

ggs_effective(fert.ggs)

Ratios all very high.

More diagnostics

ggs_crosscorrelation(fert.ggs)

ggs_grb(fert.ggs)

brms

bayesplot

The bayesplot package offers a range of MCMC diagnostics as well as Posterior Probability Checks (PPC), all of which have a convenient plot() interface. Lets start with the MCMC diagnostics.

available_mcmc()

## bayesplot MCMC module:
##   mcmc_acf
##   mcmc_acf_bar
##   mcmc_areas
##   mcmc_areas_data
##   mcmc_areas_ridges
##   mcmc_areas_ridges_data
##   mcmc_combo
##   mcmc_dens
##   mcmc_dens_chains
##   mcmc_dens_chains_data
##   mcmc_dens_overlay
##   mcmc_hex
##   mcmc_hist
##   mcmc_hist_by_chain
##   mcmc_intervals
##   mcmc_intervals_data
##   mcmc_neff
##   mcmc_neff_data
##   mcmc_neff_hist
##   mcmc_nuts_acceptance
##   mcmc_nuts_divergence
##   mcmc_nuts_energy
##   mcmc_nuts_stepsize
##   mcmc_nuts_treedepth
##   mcmc_pairs
##   mcmc_parcoord
##   mcmc_parcoord_data
##   mcmc_rank_hist
##   mcmc_rank_overlay
##   mcmc_recover_hist
##   mcmc_recover_intervals
##   mcmc_recover_scatter
##   mcmc_rhat
##   mcmc_rhat_data
##   mcmc_rhat_hist
##   mcmc_scatter
##   mcmc_trace
##   mcmc_trace_data
##   mcmc_trace_highlight
##   mcmc_violin

Of these, we will focus on:

• trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different shade of blue, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).

fert.brm3 %>% mcmc_plot(type = "combo")

fert.brm3 %>% mcmc_plot(type = "trace")

fert.brm3 %>% mcmc_plot(type = "dens_overlay")

The chains appear well mixed and very similar

• acf_bar (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain

fert.brm3 %>% mcmc_plot(type = "acf_bar")

There is no evidence of autocorrelation in the MCMC samples

• rhat_hist: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.

fert.brm3 %>% mcmc_plot(type = "rhat_hist")

All Rhat values are below 1.05, suggesting the chains have converged.

• neff_hist (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

  If the ratios are low, tightening the priors may help.

fert.brm3 %>% mcmc_plot(type = "neff_hist")

Ratios all very high.

More diagnostics

fert.brm3 %>% mcmc_plot(type = "combo")

fert.brm3 %>% mcmc_plot(type = "violin")

stan plots

The rstan package offers a range of MCMC diagnostics. Lets start with the MCMC diagnostics.

Of these, we will focus on:

• stan_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).

fert.brm3$fit %>% stan_trace()

fert.brm3$fit %>% stan_trace(inc_warmup = TRUE)

The chains appear well mixed and very similar

• stan_acf (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain

fert.brm3$fit %>% stan_ac()

There is no evidence of autocorrelation in the MCMC samples

• stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.

fert.brm3$fit %>% stan_rhat()

All Rhat values are below 1.05, suggesting the chains have converged.

• stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

  If the ratios are low, tightening the priors may help.

fert.brm3$fit %>% stan_ess()

Ratios all very high.

fert.brm3$fit %>% stan_dens(separate_chains = TRUE)

ggmcmc

The ggmcmc package also has a set of MCMC diagnostic functions. Lets start with the MCMC diagnostics.

Of these, we will focus on:

• ggs_traceplot: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).

fert.ggs <- fert.brm3$fit %>% ggs(inc_warmup = TRUE, burnin = TRUE) # does not seem to ignore burnin?
fert.ggs %>% ggs_traceplot()

fert.ggs <- fert.brm3$fit %>% ggs(inc_warmup = FALSE, burnin = TRUE) # does not seem to ignore burnin?
fert.ggs %>% ggs_traceplot()

The chains appear well mixed and very similar

• gss_autocorrelation (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain

fert.ggs %>% ggs_autocorrelation()

There is no evidence of autocorrelation in the MCMC samples

• stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.

fert.ggs %>% ggs_Rhat()

All Rhat values are below 1.05, suggesting the chains have converged.

• stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

  If the ratios are low, tightening the priors may help.

fert.ggs %>% ggs_effective()

Ratios all very high.

More diagnostics

fert.ggs %>% ggs_crosscorrelation()

fert.ggs %>% ggs_grb()

INLA

The sampling diagnostics are not relevant to INLA. Instead, we can focus on CPO and PIT

fert.inla1.cpo <- data.frame(
  cpo = fert.inla1$cpo$cpo,
  pit = fert.inla1$cpo$pit,
  failure = fert.inla1$cpo$failure
) %>%
  filter(failure == 0) %>%
  mutate(row = 1:n())

g1 <- fert.inla1.cpo %>%
  ggplot(aes(y = cpo, x = row)) +
  geom_point()
g2 <- fert.inla1.cpo %>%
  ggplot(aes(x = cpo)) +
  geom_histogram()
g3 <- fert.inla1.cpo %>%
  ggplot(aes(y = pit, x = row)) +
  geom_point()
g4 <- fert.inla1.cpo %>%
  ggplot(aes(x = pit)) +
  geom_histogram()

(g1 + g2) / (g3 + g4)

Ideally, we want to see that no CPO or PIT is very different in value from any others (not the case here) and that the histograms are relatively flat (which they kind of are here - particularly considering the small amount of data).

Model validation

Posterior probability checks

available_ppc()

Package	Description	function	rstanarm	brms
bayesplot	Density overlay	ppc_dens_overlay	pp_check(mod, plotfun='dens_overlay')	pp_check(mod, type='dens_overlay')
	Obs vs Pred error	ppc_error_scatter_avg	pp_check(mod, plotfun='error_scatter_avg')	pp_check(mod, type='error_scatter_avg')
	Pred error vs x	ppc_error_scatter_avg_vs_x	pp_check(mod, x=, plotfun='error_scatter_avg_vs_x')	pp_check(mod, x=, type='error_scatter_avg_vs_x')
	Preds vs x	ppc_intervals	pp_check(mod, x=, plotfun='intervals')	pp_check(mod, x=, type='intervals')
	Partial plot	ppc_ribbon	pp_check(mod, x=, plotfun='ribbon')	pp_check(mod, x=, type='ribbon')

rstanarm

pp check

Post predictive checks provide additional diagnostics about the fit of the model. Specifically, they provide a comparison between predictions drawn from the model and the observed data used to train the model.

available_ppc()

## bayesplot PPC module:
##   ppc_bars
##   ppc_bars_grouped
##   ppc_boxplot
##   ppc_data
##   ppc_dens
##   ppc_dens_overlay
##   ppc_dens_overlay_grouped
##   ppc_ecdf_overlay
##   ppc_ecdf_overlay_grouped
##   ppc_error_binned
##   ppc_error_hist
##   ppc_error_hist_grouped
##   ppc_error_scatter
##   ppc_error_scatter_avg
##   ppc_error_scatter_avg_vs_x
##   ppc_freqpoly
##   ppc_freqpoly_grouped
##   ppc_hist
##   ppc_intervals
##   ppc_intervals_data
##   ppc_intervals_grouped
##   ppc_km_overlay
##   ppc_loo_intervals
##   ppc_loo_pit
##   ppc_loo_pit_data
##   ppc_loo_pit_overlay
##   ppc_loo_pit_qq
##   ppc_loo_ribbon
##   ppc_ribbon
##   ppc_ribbon_data
##   ppc_ribbon_grouped
##   ppc_rootogram
##   ppc_scatter
##   ppc_scatter_avg
##   ppc_scatter_avg_grouped
##   ppc_stat
##   ppc_stat_2d
##   ppc_stat_freqpoly_grouped
##   ppc_stat_grouped
##   ppc_violin_grouped

• dens_overlay: plots the density distribution of the observed data (black line) overlayed on top of 50 density distributions generated from draws from the model (light blue). Ideally, the 50 realisations should be roughly consistent with the observed data.

pp_check(fert.rstanarm3, plotfun = "dens_overlay")

The model draws appear to be consistent with the observed data.

• error_scatter_avg: this plots the observed values against the average residuals. Similar to a residual plot, we do not want to see any patterns in this plot

pp_check(fert.rstanarm3, plotfun = "error_scatter_avg")

There is no obvious pattern in the residuals.

• error_scatter_avg_vs_x: this is similar to a regular residual plot and as such should be interpreted as such.

pp_check(fert.rstanarm3, x = fert$FERTILIZER, plotfun = "error_scatter_avg_vs_x")

• intervals: plots the observed data overlayed onto of posterior predictions associated with each level of the predictor. Ideally, the observed data should all fall within the predictive intervals.

pp_check(fert.rstanarm3, x = fert$FERTILIZER, plotfun = "intervals")

• ribbon: this is just an alternative way of expressing the above plot.

pp_check(fert.rstanarm3, x = fert$FERTILIZER, plotfun = "ribbon")

The shinystan package allows the full suite of MCMC diagnostics and posterior predictive checks to be accessed via a web interface.

# library(shinystan)
# launch_shinystan(fert.rstanarm3)

DHARMa residuals

DHARMa residuals provide very useful diagnostics. Unfortunately, we cannot directly use the simulateResiduals() function to generate the simulated residuals. However, if we are willing to calculate some of the components yourself, we can still obtain the simulated residuals from the fitted stan model.

We need to supply:

• simulated (predicted) responses associated with each observation.
• observed values
• fitted (predicted) responses (averaged) associated with each observation

preds <- posterior_predict(fert.rstanarm3, nsamples = 250, summary = FALSE)
fert.resids <- createDHARMa(
  simulatedResponse = t(preds),
  observedResponse = fert$YIELD,
  fittedPredictedResponse = apply(preds, 2, median),
  integerResponse = "gaussian"
)
plot(fert.resids)

Conclusions:

• the simulated residuals do not suggest any issues with the residuals
• there is no evidence of a lack of fit.

brms

pp check

Post predictive checks provide additional diagnostics about the fit of the model. Specifically, they provide a comparison between predictions drawn from the model and the observed data used to train the model.

available_ppc()

## bayesplot PPC module:
##   ppc_bars
##   ppc_bars_grouped
##   ppc_boxplot
##   ppc_data
##   ppc_dens
##   ppc_dens_overlay
##   ppc_dens_overlay_grouped
##   ppc_ecdf_overlay
##   ppc_ecdf_overlay_grouped
##   ppc_error_binned
##   ppc_error_hist
##   ppc_error_hist_grouped
##   ppc_error_scatter
##   ppc_error_scatter_avg
##   ppc_error_scatter_avg_vs_x
##   ppc_freqpoly
##   ppc_freqpoly_grouped
##   ppc_hist
##   ppc_intervals
##   ppc_intervals_data
##   ppc_intervals_grouped
##   ppc_km_overlay
##   ppc_loo_intervals
##   ppc_loo_pit
##   ppc_loo_pit_data
##   ppc_loo_pit_overlay
##   ppc_loo_pit_qq
##   ppc_loo_ribbon
##   ppc_ribbon
##   ppc_ribbon_data
##   ppc_ribbon_grouped
##   ppc_rootogram
##   ppc_scatter
##   ppc_scatter_avg
##   ppc_scatter_avg_grouped
##   ppc_stat
##   ppc_stat_2d
##   ppc_stat_freqpoly_grouped
##   ppc_stat_grouped
##   ppc_violin_grouped

• dens_overlay: plots the density distribution of the observed data (black line) overlayed on top of 50 density distributions generated from draws from the model (light blue). Ideally, the 50 realisations should be roughly consistent with the observed data.

fert.brm3 %>% pp_check(type = "dens_overlay", nsamples = 100)

The model draws appear to be consistent with the observed data.

• error_scatter_avg: this plots the observed values against the average residuals. Similar to a residual plot, we do not want to see any patterns in this plot

fert.brm3 %>% pp_check(type = "error_scatter_avg")

There is no obvious pattern in the residuals.

• error_scatter_avg_vs_x: this is similar to a regular residual plot and as such should be interpreted as such.

fert.brm3 %>% pp_check(x = "FERTILIZER", type = "error_scatter_avg_vs_x")

• intervals: plots the observed data overlayed on top of posterior predictions associated with each level of the predictor. Ideally, the observed data should all fall within the predictive intervals.

fert.brm3 %>% pp_check(x = "FERTILIZER", type = "intervals")

• ribbon: this is just an alternative way of expressing the above plot.

pp_check(fert.brm3, x = "FERTILIZER", type = "ribbon")

The shinystan package allows the full suite of MCMC diagnostics and posterior predictive checks to be accessed via a web interface.

# library(shinystan)
# launch_shinystan(fert.brm3)

DHARMa residuals

DHARMa residuals provide very useful diagnostics. Unfortunately, we cannot directly use the simulateResiduals() function to generate the simulated residuals. However, if we are willing to calculate some of the components yourself, we can still obtain the simulated residuals from the fitted stan model.

We need to supply:

• simulated (predicted) responses associated with each observation.
• observed values
• fitted (predicted) responses (averaged) associated with each observation

preds <- fert.brm3 %>% posterior_predict(nsamples = 250, summary = FALSE)
fert.resids <- createDHARMa(
  simulatedResponse = t(preds),
  observedResponse = fert$YIELD,
  fittedPredictedResponse = apply(preds, 2, median),
  integerResponse = FALSE
)
fert.resids %>% plot()

The integerResponse argument indicates that noise will be added to the residuals such that they will become integers. This is important in for Poisson, Negative Binomial and Binomial models.

Conclusions:

• the simulated residuals do not suggest any issues with the residuals
• there is no evidence of a lack of fit.

INLA

Out-of-sample performance

INLA includes two measures of out-of-sample performace (measures of fit).

• Conditional Predictive Ordinate (CPO) is the density of the observed value of \(y_i\) within the out-of-sample (\(y−i\)) posterior predictive distribution. A small CPO value associated with an observation suggests that this observation is unlikely (or surprising) in light of the model, priors and other data in the model. In addition, the sum of the CPO values (or alternatively, the negative of the mean natural logarithm of the CPO values) is a measure of fit.

  fert.inla1$cpo$cpo

  ##  [1] 0.012054491 0.012916051 0.006822433 0.008804804 0.018153179 0.018355621
  ##  [7] 0.014801708 0.002798478 0.006057419 0.014733889

  sum(fert.inla1$cpo$cop)

  ## [1] 0

  -mean(log(fert.inla1$cpo$cpo))

  ## [1] 4.59101

  In this case there are no CPO values that are considerably smaller (order of magnitude smaller) than the others - so with respect to the model, none of the observed values would be considered ‘surprising’. Various assumptions are made behind INLA’s computations. If any of these assumptions fail for an observation, it is flagged within (in this case) cpo$failure as a non-zero. When this is the case, it is prudent to recalculate the CPO values after removing the observations associated with failure flags not equal to zero and re-fitting the model. This can be performed using the inla.cpo() function.

  fert.inla1$cpo$failure

  ##  [1] 0 0 0 0 0 0 0 0 0 0

  In this case, there are no non-zero CPO values.

• Probability Integral Transforms (PIT) provides a version of CPO that is calibrated to the level of the Gaussian field so that it is clearer whether or not any of the values are ‘small’ (all values must be between 0 and 1). A histogram of PIT values that does not look approximately uniform (flat) indicates a lack of model fit.

  Unfortunately, the following will not work inside of an Rmarkdown (but will work fine on the console)

  plot(fert.inla1,
    plot.fixed.effects = FALSE,
    plot.lincomb = FALSE,
    plot.random.effects = FALSE,
    plot.hyperparameters = FALSE,
    plot.predictor = FALSE,
    plot.q = FALSE,
    plot.cpo = TRUE,
    plot.prior = FALSE
  )

  In this case the PIT values do not appear to deviate from a uniform distribution and thus do not indicate a lack of fit.

fert.inla1.cpo <- data.frame(
  cpo = fert.inla1$cpo$cpo,
  pit = fert.inla1$cpo$pit,
  failure = fert.inla1$cpo$failure
) %>%
  filter(failure == 0) %>%
  mutate(row = 1:n())

g1 <- fert.inla1.cpo %>%
  ggplot(aes(y = cpo, x = row)) +
  geom_point()
g2 <- fert.inla1.cpo %>%
  ggplot(aes(x = cpo)) +
  geom_histogram()
g3 <- fert.inla1.cpo %>%
  ggplot(aes(y = pit, x = row)) +
  geom_point()
g4 <- fert.inla1.cpo %>%
  ggplot(aes(x = pit)) +
  geom_histogram()

(g1 + g2) / (g3 + g4)

Ideally, we want to see that no CPO or PIT is very different in value from any others (not the case here) and that the histograms are relatively flat (which they kind of are here - particularly considering the small amount of data).

In the following, the top left is the distribution of residuals and top right is the observed vs predicted values.

ggplot_inla_residuals(fert.inla1, observed = fert$YIELD)

Conclusions-

• the distribution of residuals is nearly uniform (keep in mind that the sample size is very small in this example).
• the fitted values approximate the observed values very well

The following figure represents the equivalent of a (standardised) residual plot.

ggplot_inla_residuals2(fert.inla1, observed = fert$YIELD)

DHARMa residuals

Unfortunately, DHARMa does not directly support INLA. Therefore, in order to make use of this framework, we need to do some of the heavy lifting. Specifically, we need to generate the posterior predictions associated with the observed data.

Whilst we can draw random samples from the posterior with the inla.posterior.sample() function, this will only account for uncertainty in the fixed parameter estimates. That is, it will not incorporate the overall uncertainty in the Gaussian distribution (\(\sigma\)). Therefore, we need to add this back on ourself (keep in mind that this will be in units of precision and thus needs to be back-scaled into units of standard deviation). Furthermore, we will need to use the estimates of standard deviation to estimate the random noise to add onto the predictions. This is done by resampling the family (Gaussian in this case).

I acknowledge that this (resampling the SD) appears to be a bit of a hack and seems to go against the INLA principles. Nevertheless, we are only doing this this for the purposes of checking residual diagnostics.

## draw 250 samples from the posterior
draws <- inla.posterior.sample(n = 250, result = fert.inla)
## extract the latent predictions for the observed data
preds <- t(sapply(draws, function(x) x$latent[1:nrow(fert)]))
## extract the first (family precision) hyperprior
preds.hyper <- sapply(draws, function(x) 1 / sqrt(x$hyperpar[[1]]))
## use this to generate gaussian noise
preds.hyper <- rnorm(length(preds.hyper), mean = 0, sd = preds.hyper)
## add the noise to each prediction
preds <- sweep(preds, 1, preds.hyper, FUN = "+")
## generate the DHARMa residuals
fert.resids <- createDHARMa(
  simulatedResponse = t(preds),
  observedResponse = fert$YIELD,
  fittedPredictedResponse = apply(preds, 2, median),
  integerResponse = FALSE
)
fert.resids %>% plot()

Conclusions:

• the simulated residuals do not suggest any issues with the residuals
• there is no evidence of a lack of fit.

Partial effects plots

rstanarm

ggpredict

fert.rstanarm3 %>%
  ggpredict() %>%
  plot(add.data = TRUE)

## $FERTILIZER

ggemmeans

fert.rstanarm3 %>%
  ggemmeans(~FERTILIZER) %>%
  plot(add.data = TRUE)

fitted_draws

fert.rstanarm3 %>%
  fitted_draws(newdata = fert) %>%
  median_hdci() %>%
  ggplot(aes(x = FERTILIZER, y = .value)) +
  geom_ribbon(aes(ymin = .lower, ymax = .upper), fill = "blue", alpha = 0.3) +
  geom_line()

brms

conditional_effects

fert.brm3 %>%
  ggemmeans(~FERTILIZER) %>%
  plot(add.data = TRUE)

fert.brm3 %>% conditional_effects()

fert.brm3 %>%
  conditional_effects() %>%
  plot(points = TRUE)

fert.brm3 %>%
  conditional_effects(spaghetti = TRUE, nsamples = 200) %>%
  plot(points = TRUE)

ggpredict

fert.brm3 %>%
  ggpredict() %>%
  plot(add.data = TRUE)

## $FERTILIZER

ggemmeans

fert.brm3 %>%
  ggemmeans(~FERTILIZER) %>%
  plot(add.data = TRUE)

fitted_draws

fert.brm3 %>%
  fitted_draws(newdata = fert) %>%
  median_hdci() %>%
  ggplot(aes(x = FERTILIZER, y = .value)) +
  geom_ribbon(aes(ymin = .lower, ymax = .upper), fill = "blue", alpha = 0.3) +
  geom_line()

INLA

plot(fert.inla1,
  plot.fixed.effects = FALSE,
  plot.lincomb = FALSE,
  plot.random.effects = FALSE,
  plot.hyperparameters = FALSE,
  plot.predictor = TRUE,
  plot.q = FALSE,
  plot.cpo = FALSE,
  plot.prior = FALSE
)

newdata <- fert %>% tidyr::expand(FERTILIZER = modelr::seq_range(FERTILIZER, n = 100))
Xmat <- model.matrix(~FERTILIZER, newdata)

nms <- colnames(fert.inla1$model.matrix)
n <- sapply(nms, function(x) 0, simplify = FALSE)
draws <- inla.posterior.sample(n = 250, result = fert.inla1, selection = n)
coefs <- t(sapply(draws, function(x) x$latent))
Fit <- coefs %*% t(Xmat) %>%
  as.data.frame() %>%
  mutate(Rep = 1:n()) %>%
  pivot_longer(cols = -Rep) %>%
  group_by(name) %>%
  median_hdci(value) %>%
  ungroup() %>%
  mutate(name = as.integer(as.character(name))) %>%
  arrange(name)
newdata <- newdata %>%
  bind_cols(Fit)

newdata %>%
  ggplot(aes(x = FERTILIZER)) +
  geom_ribbon(aes(ymin = .lower, ymax = .upper), fill = "orange", color = NA, alpha = 0.3) +
  geom_line(aes(y = value), color = "orange") +
  geom_point(data = fert, aes(y = YIELD)) +
  theme_classic()

Model investigation

Rather than simply return point estimates of each of the model parameters, Bayesian analyses capture the full posterior of each parameter. These are typically stored within the list structure of the output object.

As with most statistical routines, the overloaded summary() function provides an overall summary of the model parameters. Typically, the summaries will include the means / medians along with credibility intervals and perhaps convergence diagnostics (such as R hat). However, more thorough investigation and analysis of the parameter posteriors requires access to the full posteriors.

There is currently a plethora of functions for extracting the full posteriors from models. In part, this is a reflection of a rapidly evolving space with numerous packages providing near equivalent functionality (it should also be noted, that over time, many of the functions have been deprecated due to inconsistencies in their names). Broadly speaking, the functions focus on draws from the posterior of either the parameters (intercept, slope, standard deviation etc), linear predictor, expected values or predicted values. The distinction between the latter three are highlighted in the following table.

Property	Description
linear predictors	values predicted on the link scale
expected values	predictions (on response scale) without residual error (predicting expected mean outcome(s))
predicted values	predictions (on response scale) that incorporate residual error
fitted values	predictions on the response scale

The following table lists the various ways of extracting the full posteriors of the model parameters parameters, expected values and predicted values. The crossed out items are now deprecated and function with a namespace of __ mean that the functionality is provided via a range of packages.

Function	Values	Description
__::as.matrix()	Parameters	Returns \(n\times p\) matrix
__::as.data.frame()	Parameters	Returns \(n\times p\) data.frame
__::as_tibble()	Parameters	Returns \(n\times p\) tibble
posterior::as_draws_df()	Parameters	Returns \(n\times p\) data.frame with additional info about chain, interaction and draw
brms::posterior_samples()	Parameters	Returns \(n\times p\) data.frame
tidybayes::tidy_draws()	Parameters	Returns \(n\times p\) tibble with addition info about the chain, iteration and draw
rstan::extract()	Parameters	Returns a \(p\) length list of \(n\) length vectors
tidybayes::spread_draws()	Parameters	Returns \(n\times r\) tibble with additional info about chain, interaction and draw
tidybayes::gather_draws()	Parameters	Returns a gathered spread_draws tibble with additional info about chain, interaction and draw
rstanarm::posterior_linpred()	Linear predictors	Returns \(n\times N\) tibble on the link scale
brms::posterior_linpred()	Linear predictors	Returns \(n\times N\) tibble on the link scale
tidybayes::linpred_draws()	Linear predictors	Returns tibble with $nN rows and .linpred on the link scale additional info about chain, interaction and draw
rstanarm::posterior_epred()	Expected values	Returns \(n\times N\) tibble on the response scale
brms::posterior_epred()	Expected values	Returns \(n\times N\) tibble on the response scale
tidybayes::epred_draws()	Expected values	Returns tibble with $nN rows and .epred on the response scale additional info about chain, interaction and draw
rstanarm::posterior_predict()	Expected values	Returns \(n\times N\) tibble of predictions (including residuals) on the response scale
brms::posterior_predict()	Expected values	Returns \(n\times N\) tibble of predictions (including residuals) on the response scale
tidybayes::predicted_draws()	Expected values	Returns tibble with $nN rows and .prediction (including residuals) on the response scale additional info about chain, interaction and draw

where \(n\) is the number of MCMC samples and \(p\) is the number of parameters to estimate, \(N\) is the number of newdata rows and \(r\) is the number of requested parameters. For the tidybayes versions in the table above, the function expects a model to be the first parameter (and a dataframe to be the second). There are also add_ versions which expect a dataframe to be the first argument and the model to be the second. These alternatives facilitate pipings with different starting objects.

Function	Description
median_qi	Median and quantiles of specific columns
median_hdi	Median and Highest Probability Density Interval of specific columns
median_hdci	Median and continuous Highest Probability Density Interval of specific columns
tidyMCMC	Median/mean and quantiles/hpd of all columns

It is important to remember that when working with Bayesian models, everything is a distribution. Whilst point estimates (such as a mean) of the parameters can be calculated from these distributions, we start off with a large number of estimates per parameter.

rstanarm

rstanarm captures the MCMC samples from stan within the returned list. There are numerous ways to retrieve and summarise these samples. The first three provide convenient numeric summaries from which you can draw conclusions, the last four provide ways of obtaining the full posteriors.

summary

The summary() method generates simple summaries (mean, standard deviation as well as 10, 50 and 90 percentiles).

fert.rstanarm3 %>% summary()

## 
## Model Info:
##  function:     stan_glm
##  family:       gaussian [identity]
##  formula:      YIELD ~ FERTILIZER
##  algorithm:    sampling
##  sample:       2400 (posterior sample size)
##  priors:       see help('prior_summary')
##  observations: 10
##  predictors:   2
## 
## Estimates:
##               mean   sd   10%   50%   90%
## (Intercept) 52.7   14.0 36.1  52.4  69.5 
## FERTILIZER   0.8    0.1  0.7   0.8   0.9 
## sigma       19.6    5.3 14.3  18.7  26.0 
## 
## Fit Diagnostics:
##            mean   sd    10%   50%   90%
## mean_PPD 163.4    9.1 152.0 163.5 174.5
## 
## The mean_ppd is the sample average posterior predictive distribution of the outcome variable (for details see help('summary.stanreg')).
## 
## MCMC diagnostics
##               mcse Rhat n_eff
## (Intercept)   0.3  1.0  2036 
## FERTILIZER    0.0  1.0  2182 
## sigma         0.1  1.0  2303 
## mean_PPD      0.2  1.0  2422 
## log-posterior 0.0  1.0  2178 
## 
## For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).

Conclusions:

• in the Model Info, we are informed that the total MCMC posterior sample size is 2400 and that there were 10 raw observations.
• the estimated intercept (expected yield when fertilizer concentration is 0) is 52.67. This is the mean of the posterior distribution for this parameter.
• the estimated slope (rate at which yield changes per 1 unit change in fertilizer concentration), is 0.81 (mean) or 0.69 (median) with a standard deviation of 0. The 90% credibility intervals indicate that we are 90% confident that the slope is between 0.81 and 0.81 - e.g. there is a significant positive trend.
• sigma is estimated to be 19.62
• Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.

tidyMCMC

tidyMCMC(fert.rstanarm3$stanfit,
  estimate.method = "median", conf.int = TRUE,
  conf.method = "HPDinterval", rhat = TRUE, ess = TRUE
)

Conclusions:

• the estimated intercept (expected yield when fertilizer concentration is 0) is 52.41. This is the median of the posterior distribution for this parameter.
• the estimated slope (rate at which yield changes per 1 unit change in fertilizer concentration), is 0.81 (median) with a standard error of 0.09. The 95% credibility intervals indicate that we are 95% confident that the slope is between 0.63 and 0.98 - e.g. there is a significant positive trend.
• sigma is estimated to be 18.74
• Rhat and number of effective samples for each parameter are also provided and all look good.

as_draws_df (posterior)

fert.rstanarm3$stanfit %>% as_draws_df()

fert.rstanarm3$stanfit %>%
  as_draws_df() %>%
  summarise_draws(
    "median",
    ~ HDInterval::hdi(.x),
    "rhat",
    "ess_bulk"
  )

tidy_draws

This is purely a graphical depiction on the posteriors.

fert.rstanarm3 %>% tidy_draws()

gather_draws

fert.draw <- fert.rstanarm3 %>% gather_draws(`(Intercept)`, FERTILIZER, sigma)
## OR via regex
fert.draw <- fert.rstanarm3 %>% gather_draws(`.Intercept.*|FERT.*|sigma`, regex = TRUE)
fert.draw

We can then summarise this

fert.draw %>% median_hdci()

fert.rstanarm3 %>%
  gather_draws(`(Intercept)`, FERTILIZER, sigma) %>%
  ggplot() +
  stat_halfeye(aes(x = .value, y = .variable)) +
  facet_wrap(~.variable, scales = "free")

Conclusions:

• the estimated intercept (expected yield when fertilizer concentration is 0) is 52.41. This is the median of the posterior distribution for this parameter.
• the estimated slope (rate at which yield changes per 1 unit change in fertilizer concentration), is 0.81 (mean) or (median). The 95% credibility intervals indicate that we are 95% confident that the slope is between 0.63 and 0.98 - e.g. there is a significant positive trend.
• sigma is estimated to be 18.74

bayesplot

fert.rstanarm3 %>% plot(plotfun = "mcmc_intervals")

spread_draws

fert.rstanarm3 %>% spread_draws(`(Intercept)`, FERTILIZER, sigma)

# OR via regex
fert.rstanarm3 %>% spread_draws(`.Intercept.*|FERT.*|sigma`, regex = TRUE)

posterior_samples

Note, this is not deprecated

fert.rstanarm3 %>%
  posterior_samples() %>%
  as_tibble()

\(R^2\)

fert.rstanarm3 %>%
  bayes_R2() %>%
  median_hdci()

Empirical p-value

It is possible to obtain an empirical p-value (for those that cannot live without p-values). This is essentially compares (for any posterior) that the mean is zero compared to a multivariate distribution with elliptical contours.

mcmcpvalue <- function(samp) {
  ## elementary version that creates an empirical p-value for the
  ## hypothesis that the columns of samp have mean zero versus a
  ## general multivariate distribution with elliptical contours.

  ## differences from the mean standardized by the observed
  ## variance-covariance factor

  ## Note, I put in the bit for single terms
  if (length(dim(samp)) == 0) {
    std <- backsolve(chol(var(samp)), cbind(0, t(samp)) - mean(samp), transpose = TRUE)
    sqdist <- colSums(std * std)
    sum(sqdist[-1] > sqdist[1]) / length(samp)
  } else {
    std <- backsolve(chol(var(samp)), cbind(0, t(samp)) - colMeans(samp), transpose = TRUE)
    sqdist <- colSums(std * std)
    sum(sqdist[-1] > sqdist[1]) / nrow(samp)
  }
}

mcmcpvalue(as.matrix(fert.rstanarm3)[, c("FERTILIZER")])

## [1] 0

brms

brms captures the MCMC samples from stan within the returned list. There are numerous ways to retrieve and summarise these samples. The first three provide convenient numeric summaries from which you can draw conclusions, the last four provide ways of obtaining the full posteriors.

summary

The summary() method generates simple summaries (mean, standard deviation as well as 10, 50 and 90 percentiles).

fert.brm3 %>% summary()

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: YIELD ~ FERTILIZER 
##    Data: fert (Number of observations: 10) 
##   Draws: 3 chains, each with iter = 1000; warmup = 200; thin = 5;
##          total post-warmup draws = 480
## 
## Population-Level Effects: 
##            Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept     52.46      8.81    35.87    70.64 1.00     2497     2367
## FERTILIZER     0.81      0.06     0.69     0.92 1.00     2437     2344
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    12.71      1.81     9.65    16.88 1.00     2409     2256
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Conclusions:

• in the initial block of information, we are reminded on the formula as well as chain fitting characteristics. We are also informed that the total number of post-warmup MCMC samples is is 2400.
• the estimated intercept (expected yield when fertilizer concentration is 0) is 52.46. This is the mean of the posterior distribution for this parameter.
• the estimated slope (rate at which yield changes per 1 unit change in fertilizer concentration), is 0.81 (mean) with a standard error of 0.06. The 95% credibility intervals indicate that we are 95% confident that the slope is between 0.69 and 0.92 - e.g. there is a significant positive trend.
• sigma is estimated to be 12.71
• Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.

tidyMCMC

fert.brm3$fit %>%
  tidyMCMC(
    estimate.method = "median",
    conf.int = TRUE,
    conf.method = "HPDinterval",
    rhat = TRUE, ess = TRUE
  )

Conclusions:

• the estimated intercept (expected yield when fertilizer concentration is 0) is 52.23. This is the median of the posterior distribution for this parameter.
• the estimated slope (rate at which yield changes per 1 unit change in fertilizer concentration), is 0.81 (median) with a standard error of 0.06. The 95% credibility intervals indicate that we are 95% confident that the slope is between 0.69 and 0.91 - e.g. there is a significant positive trend.
• sigma is estimated to be 12.55
• Rhat and number of effective samples for each parameter are also provided and all look good.

as_draws_df (posterior)

fert.brm3 %>% as_draws_df()

tidy_draws

Return the draws (samples) for each parameter in wide format

fert.brm3 %>% tidy_draws()

The above can be useful for exploring the full posteriors of all the parameters of performing specific calculations using the posteriors, summarising the parameters takes a few more steps.

fert.brm3 %>%
  tidy_draws() %>%
  gather_variables() %>%
  median_hdci()

The gather_draws on the other hand, conveniently combines tidy_draws and gather_variables together in a single command. Furthermore, it returns all of the variables. The spread_draws function allows users to target specific variables (either by naming them in full or via regexp).

gather_draws

fert.brm3 %>% get_variables()

##  [1] "b_Intercept"     "b_FERTILIZER"    "sigma"           "prior_Intercept"
##  [5] "prior_b"         "prior_sigma"     "lp__"            "accept_stat__"  
##  [9] "stepsize__"      "treedepth__"     "n_leapfrog__"    "divergent__"    
## [13] "energy__"

fert.brm3 %>%
  gather_draws(b_Intercept, b_FERTILIZER, sigma) %>%
  median_hdci()

## OR via regex
fert.brm3 %>%
  gather_draws(`b_.*|sigma`, regex = TRUE) %>%
  median_hdci()

fert.brm3 %>%
  gather_draws(b_Intercept, b_FERTILIZER, sigma) %>%
  ggplot() +
  stat_halfeye(aes(x = .value, y = .variable)) +
  facet_wrap(~.variable, scales = "free")

fert.brm3 %>%
  gather_draws(b_Intercept, b_FERTILIZER, sigma) %>%
  ggplot() +
  stat_halfeye(aes(x = .value, y = .variable))

Conclusions:

• the estimated intercept (expected yield when fertilizer concentration is 0) is 0.81. This is the median of the posterior distribution for this parameter.
• the estimated slope (rate at which yield changes per 1 unit change in fertilizer concentration), is 52.23 (mean) or (median). The 95% credibility intervals indicate that we are 95% confident that the slope is between 35.76 and 70.44 - e.g. there is a significant positive trend.
• sigma is estimated to be 12.55

spread_draws

Select just the parameters of interest.

fert.brm3 %>% spread_draws(b_Intercept, b_FERTILIZER, sigma)

# OR via regex
fert.brm3 %>% spread_draws(`b_.*|sigma`, regex = TRUE)

bayesplot

fert.brm3 %>% mcmc_plot(type = "intervals")

posterior_samples

Note, this is now deprecated

fert.brm3 %>%
  posterior_samples() %>%
  as_tibble()

\(R^2\)

fert.brm3 %>%
  bayes_R2(summary = FALSE) %>%
  median_hdci()

Empirical p-value

It is possible to obtain an empirical p-value (for those that cannot live without p-values). This is essentially compares (for any posterior) that the mean is zero compared to a multivariate distribution with elliptical contours.

mcmcpvalue <- function(samp) {
  ## elementary version that creates an empirical p-value for the
  ## hypothesis that the columns of samp have mean zero versus a
  ## general multivariate distribution with elliptical contours.

  ## differences from the mean standardized by the observed
  ## variance-covariance factor

  ## Note, I put in the bit for single terms
  if (length(dim(samp)) == 0) {
    std <- backsolve(chol(var(samp)), cbind(0, t(samp)) - mean(samp), transpose = TRUE)
    sqdist <- colSums(std * std)
    sum(sqdist[-1] > sqdist[1]) / length(samp)
  } else {
    std <- backsolve(chol(var(samp)), cbind(0, t(samp)) - colMeans(samp), transpose = TRUE)
    sqdist <- colSums(std * std)
    sum(sqdist[-1] > sqdist[1]) / nrow(samp)
  }
}

mcmcpvalue(as.matrix(fert.brm3)[, c("b_FERTILIZER")])

## [1] 0

INLA

INLA captures a summary of the parameter posteriors within the fitted object.

summary

The summary() method generates simple summaries (mean, standard deviation as well as 10, 50 and 90 percentiles).

fert.inla1 %>% summary()

## 
## Call:
##    c("inla(formula = YIELD ~ FERTILIZER, family = \"gaussian\", data = 
##    fert, ", " control.compute = list(config = TRUE, dic = TRUE, waic = 
##    TRUE, ", " cpo = TRUE), control.family = list(hyper = list(prec = 
##    list(prior = \"loggamma\", ", " param = c(0.5, 0.31)))), control.fixed 
##    = list(mean.intercept = 80, ", " prec.intercept = 1e-05, mean = 0, prec 
##    = 0.0384))") 
## Time used:
##     Pre = 0.946, Running = 0.106, Post = 0.04, Total = 1.09 
## Fixed effects:
##               mean     sd 0.025quant 0.5quant 0.975quant   mode kld
## (Intercept) 52.023 13.981     24.003   52.011     80.068 51.996   0
## FERTILIZER   0.811  0.090      0.630    0.811      0.991  0.811   0
## 
## Model hyperparameters:
##                                          mean    sd 0.025quant 0.5quant
## Precision for the Gaussian observations 0.003 0.001      0.001    0.003
##                                         0.975quant  mode
## Precision for the Gaussian observations      0.007 0.002
## 
## Expected number of effective parameters(stdev): 2.00(0.002)
## Number of equivalent replicates : 5.01 
## 
## Deviance Information Criterion (DIC) ...............: 91.62
## Deviance Information Criterion (DIC, saturated) ....: 14.69
## Effective number of parameters .....................: 3.21
## 
## Watanabe-Akaike information criterion (WAIC) ...: 91.36
## Effective number of parameters .................: 2.49
## 
## Marginal log-Likelihood:  -54.83 
## CPO and PIT are computed
## 
## Posterior marginals for the linear predictor and
##  the fitted values are computed

Conclusions:

• in the initial block of information, we are reminded of the function call
• a breakdown of the various components of the time taken for the analysis to run
• the ‘Fixed effects’ indicate that the:
  • estimated Intercept is 52.023 (mean) or 52.011 (median)
  • estimated slope is 0.811 (mean) or 0.811 (median)
  • the 95% Credibility intervals indicate that we are 95% confident that the slope is between 0.63 and 0.991
• the `Model hyperparameters’ indicate that the:
  • the precision of the Gaussian distribution is 0.003 (mean) and 0.003 (median) . Precision is \(\tau = 1/\sigma^2\) and thus, \(\sigma\) is estimated to be 18.2574186
  • WAIC (Watanabe-Akaike information criterion is estimated to be 91.364

Plot of posteriors

Fixed effects

brinla::bri.fixed.plot(fert.inla1)

Hyperparameters

brinla::bri.hyperpar.plot(fert.inla1)

Posterior draws

Since INLA is a Bayesian approximation, unlike full Bayesian methods, there are not chains of posterior draws. However, we can use the inla.posterior.sample() function, to generate draws of parameter and fitted value posteriors from the fitted model. In the following code snippet, we will generate 1000 draws from the fitted model (for repeatability, I will also set a random seed).

The output of this function is a list and items are packed in according to the sample (draw) number. I find this awkward to work with and prefer to have the data organised by parameters. However, prior to reorganising the list, I will query the latent (fixed and fitted values) parameter names to help with filtering to just the ones I am interested in later on.

## Get all the draws as a list
n <- 1000
draws <- inla.posterior.sample(n = n, result = fert.inla1, seed = 1)
## Redimension the list for the latent (fixed and fitted) values
(latent.names <- draws[[1]]$latent %>% rownames())

##  [1] "Predictor:1"   "Predictor:2"   "Predictor:3"   "Predictor:4"  
##  [5] "Predictor:5"   "Predictor:6"   "Predictor:7"   "Predictor:8"  
##  [9] "Predictor:9"   "Predictor:10"  "(Intercept):1" "FERTILIZER:1"

The names that begin with Predictor: are the fitted values. Essentially, they are the estimated predicted values associated with the observed data. The (Intercept):1 and FERTILIZER:1 are the intercept and slope respectively.

Now to convert the list of draws into a data.frame.

draws <- sapply(draws, function(x) x[["latent"]]) %>%
  as.data.frame() %>%
  set_names(paste0("Draw", 1:n)) %>%
  mutate(Parameter = latent.names) %>%
  dplyr::select(Parameter, everything())
draws

This will convert the list into a \(s\times p\) data.frame, where \(s\) is the number of draws (1000 in this case) and \(p\) is the number of latent parameters (12 in this case) plus an additional column to keep track of the Parameters.

Finally, we might like to lengthen this data set for more convenient plotting and summarising.

fert.draws <- draws %>% pivot_longer(cols = -Parameter, names_to = "Draws")
fert.draws

Intervals

To focus on and filter to just the fixed effects, we could now:

fert.draws %>%
  filter(grepl("Intercept|FERTILIZER", Parameter)) %>%
  group_by(Parameter) %>%
  median_hdci(value)

fert.draws %>%
  filter(grepl("Intercept|FERTILIZER", Parameter)) %>%
  ggplot(aes(x = value, y = Parameter)) +
  geom_vline(xintercept = 0, linetype = "dashed") +
  stat_pointinterval(point_interval = median_hdci)

## or separate into panels
fert.draws %>%
  filter(grepl("Intercept|FERTILIZER", Parameter)) %>%
  ggplot(aes(x = value)) +
  geom_vline(xintercept = 0) +
  stat_pointinterval(point_interval = median_hdci) +
  facet_grid(~Parameter, scales = "free_x")

Density intervals

fert.draws %>%
  filter(grepl("Intercept|FERTILIZER", Parameter)) %>%
  group_by(Parameter) %>%
  nest() %>%
  mutate(
    hpd = map(data, ~ median_hdci(.x$value)),
    Density = map(data, function(.x) {
      d <- density(.x$value)
      data.frame(x = d$x, y = d$y)
    }),
    Quant = map2(Density, hpd, ~ factor(findInterval(.x$x, .y[, c("ymin", "ymax")])))
  ) %>%
  unnest(c(Density, Quant)) %>%
  ggplot(aes(x = x, y = y, fill = Quant)) +
  geom_vline(xintercept = 0) +
  geom_ribbon(aes(ymin = 0, ymax = y)) +
  facet_wrap(~Parameter, scales = "free")

Ridges

fert.draws %>%
  filter(grepl("Intercept|FERTILIZER", Parameter)) %>%
  ggplot(aes(x = value, y = Parameter, fill = stat(quantile))) +
  geom_vline(xintercept = 0, linetype = "dashed") +
  ggridges::stat_density_ridges(
    geom = "density_ridges_gradient",
    calc_ecdf = TRUE,
    quantile_fun = function(x, probs) quantile(x, probs),
    quantiles = c(0.025, 0.975)
  ) +
  scale_fill_viridis_d()

fert.draws %>%
  filter(grepl("Intercept|FERTILIZER", Parameter)) %>%
  ggplot(aes(x = value, y = Parameter, fill = stat(quantile))) +
  geom_vline(xintercept = 0, linetype = "dashed") +
  ggridges::stat_density_ridges(
    geom = "density_ridges_gradient",
    calc_ecdf = TRUE,
    quantile_fun = function(x, probs) quantile(x, probs),
    quantiles = c(0.025, 0.975)
  ) +
  scale_fill_viridis_d() +
  facet_grid(~Parameter, scales = "free_x")

\(R^2\)

cor(
  fert.inla1$summary.fitted.values[, "mean"],
  fert$YIELD
)^2

## [1] 0.9216005

Predictions

There are a large number of candidate routines for performing prediction. We will go through many of these. It is worth noting that prediction is technically the act of estimating what we expect to get if we were to collect a single new observation from a particular population (e.g. a specific level of fertilizer concentration). Often this is not what we want. Often we want the fitted values - estimates of what we expect to get if we were to collect multiple new observations and average them.

So while fitted values represent the expected underlying processes occurring in the system, predicted values represent our expectations from sampling from such processes.

Package	Function	Description	Summarise with
rstantools	posterior_predict	Draw from the posterior of a prediction (includes sigma) - predicts single observations	tidyMCMC()
rstantools	posterior_linpred	Draw from the posterior of the fitted values (on the link scale) - predicts average observations	tidyMCMC()
rstantools	posterior_epred	Draw from the posterior of the fitted values (on the response scale) - predicts average observations	tidyMCMC()
tidybayes	predicted_draws	Extract the posterior of prediction values	median_hdci()
tidybayes	epred_draws	Extract the posterior of expected values	median_hdci()
tidybayes	fitted_draws	Extract the posterior of fitted values	median_hdci()
tidybayes	add_predicted_draws	Adds draws from the posterior of predictions to a data frame (of prediction data)	median_hdci()
tidybayes	add_fitted_draws	Adds draws from the posterior of fitted values to a data frame (of prediction data)	median_hdci()
emmeans	emmeans	Estimated marginal means from which posteriors can be drawn (via tidy_draws	median_hdci()

For simple models prediction is essentially taking the model formula complete with parameter (coefficient) estimates and solving for new values of the predictor. To explore this, we will use the fitted model to predict Yield for a Fertilizer concentration of 110.

We will therefore start by establishing this prediction domain as a data frame to use across all of the prediction routines.

## establish a data set that defines the new data to predict against
newdata <- data.frame(FERTILIZER = 110)

rstanarm

emmeans

fert.rstanarm3 %>% emmeans(~FERTILIZER, at = newdata)

##  FERTILIZER emmean lower.HPD upper.HPD
##         110    141       127       154
## 
## Point estimate displayed: median 
## HPD interval probability: 0.95

predicted values

fert.rstanarm3 %>%
  tidybayes::predicted_draws(newdata) %>%
  median_hdci()

## or
newdata %>%
  tidybayes::add_predicted_draws(fert.rstanarm3) %>%
  median_hdci()

## or
fert.rstanarm3 %>%
  posterior_predict(newdata = newdata) %>%
  tidyMCMC(estimate.method = "median", conf.int = TRUE, conf.method = "HPDinterval")

expected values

fert.rstanarm3 %>%
  epred_draws(newdata) %>%
  median_hdci()

## or
newdata %>%
  add_epred_draws(fert.rstanarm3) %>%
  median_hdci()

## or
fert.rstanarm3 %>%
  posterior_epred(newdata = newdata) %>%
  tidyMCMC(estimate.method = "median", conf.int = TRUE, conf.method = "HPDinterval")

linear predictor

fert.rstanarm3 %>%
  linpred_draws(newdata) %>%
  median_hdci()

## or
newdata %>%
  add_linpred_draws(fert.rstanarm3) %>%
  median_hdci()

## or
fert.rstanarm3 %>%
  posterior_linpred(newdata = newdata) %>%
  tidyMCMC(estimate.method = "median", conf.int = TRUE, conf.method = "HPDinterval")

manual calculations

## Establish an appropriate model matrix
Xmat <- model.matrix(~FERTILIZER, data = newdata)
### get the posterior draws for the linear predictor
coefs <- fert.rstanarm3$stanfit %>%
  as_draws_df() %>%
  dplyr::select(c("(Intercept)", "FERTILIZER")) %>%
  as.matrix()
fit <- coefs %*% t(Xmat)
fit %>% median_hdci()

## or
coefs <- fert.rstanarm3 %>%
  tidy_draws() %>%
  dplyr::select(c("(Intercept)", "FERTILIZER")) %>%
  as.matrix()
fit <- coefs %*% t(Xmat)
fit %>% median_hdci()

## or
coefs <- fert.rstanarm3$stanfit %>%
  as_draws_matrix() %>%
  subset_draws(c("(Intercept)", "FERTILIZER"))
fit <- coefs %*% t(Xmat)
fit %>% median_hdci()

brms

emmeans

This is the option we will focus on for most of the worksheets

fert.brm3 %>%
  emmeans(~FERTILIZER, at = newdata)

##  FERTILIZER emmean lower.HPD upper.HPD
##         110    141       133       150
## 
## Point estimate displayed: median 
## HPD interval probability: 0.95

## OR
fert.brm3 %>%
  emmeans(~FERTILIZER, at = newdata) %>%
  tidy_draws() %>%
  median_hdci()

predicted values

fert.brm3 %>%
  tidybayes::predicted_draws(newdata) %>%
  median_hdci()

## or
newdata %>%
  tidybayes::add_predicted_draws(fert.brm3) %>%
  median_hdci()

## or
fert.brm3 %>%
  posterior_predict(newdata = newdata) %>%
  as.mcmc() %>%
  tidyMCMC(estimate.method = "median", conf.int = TRUE, conf.method = "HPDinterval")

expected values

fert.brm3 %>%
  epred_draws(newdata) %>%
  median_hdci()

## or
newdata %>%
  add_epred_draws(fert.brm3) %>%
  median_hdci()

## or
fert.brm3 %>%
  posterior_epred(newdata = newdata) %>%
  as.mcmc() %>%
  tidyMCMC(estimate.method = "median", conf.int = TRUE, conf.method = "HPDinterval")

linear predictor

fert.brm3 %>%
  linpred_draws(newdata) %>%
  median_hdci()

## or
newdata %>%
  add_linpred_draws(fert.brm3) %>%
  median_hdci()

## or
fert.brm3 %>%
  posterior_linpred(newdata = newdata) %>%
  as.mcmc() %>%
  tidyMCMC(estimate.method = "median", conf.int = TRUE, conf.method = "HPDinterval")

manual calculations

## Establish an appropriate model matrix
Xmat <- model.matrix(~FERTILIZER, data = newdata)
### get the posterior draws for the linear predictor
coefs <- fert.brm3 %>%
  as_draws_df() %>%
  dplyr::select(c("b_Intercept", "b_FERTILIZER")) %>%
  as.matrix()
fit <- coefs %*% t(Xmat)
fit %>%
  median_hdci() %>%
  bind_cols(newdata)

## or
coefs <- fert.brm3 %>%
  tidy_draws() %>%
  dplyr::select(c("b_Intercept", "b_FERTILIZER")) %>%
  as.matrix()
fit <- coefs %*% t(Xmat)
fit %>%
  median_hdci() %>%
  bind_cols(newdata)

## or
coefs <- fert.brm3 %>%
  as_draws_matrix() %>%
  subset_draws(c("b_Intercept", "b_FERTILIZER"))
fit <- coefs %*% t(Xmat)
fit %>%
  median_hdci() %>%
  bind_cols(newdata)

## or
## Establish an appropriate model matrix
Xmat <- model.matrix(~FERTILIZER, data = newdata)
### get the posterior draws for the linear predictor
coefs <- fert.brm3 %>%
  posterior_samples(pars = c("b_Intercept", "b_FERTILIZER")) %>%
  as.matrix()
fit <- coefs %*% t(Xmat)
fit %>%
  median_hdci() %>%
  bind_cols(newdata)

INLA

As with most things to do with INLA, things are done a little differently. In INLA, there are three options for predictions:

1. generate draws of the parameters and calculate the outer product of these and a model matrix associated with the newdata.
2. bind the newdata to the observed data when fitting the model which will result in INLA estimating the fitted values during model fitting. In this approach, the response values associated with the newdata should all be NA.
3. define a linear combinations and pass this into the model during fitting

Lets explore each of these in turn.

From draws

## Expected values
Xmat <- model.matrix(~FERTILIZER, newdata)
nms <- colnames(fert.inla1$model.matrix)
sel <- sapply(nms, function(x) 0, simplify = FALSE)
n <- 1000
draws <- inla.posterior.sample(n = n, result = fert.inla1, selection = sel, seed = 1)
coefs <- t(sapply(draws, function(x) x$latent))
Fit <- coefs %*% t(Xmat) %>%
  as.data.frame() %>%
  mutate(Rep = 1:n()) %>%
  pivot_longer(cols = -Rep) %>%
  group_by(name) %>%
  median_hdci(value) %>%
  ungroup() %>%
  mutate(name = as.integer(as.character(name))) %>%
  arrange(name)
newdata.inla <- newdata %>%
  bind_cols(Fit)
newdata.inla

## Predictions
Fit <- coefs %*% t(Xmat)
draws <- inla.posterior.sample(n = n, result = fert.inla1, seed = 1)
sigma <- sqrt(1 / (sapply(draws, function(x) x$hyperpar)))
sigma <- sapply(sigma, function(x) rnorm(1, 0, sigma))
Fit <- sweep(Fit, MARGIN = 1, sigma, FUN = "+") %>%
  as.data.frame() %>%
  mutate(Rep = 1:n()) %>%
  pivot_longer(cols = -Rep) %>%
  group_by(name) %>%
  median_hdci(value) %>%
  bind_cols(newdata) %>%
  ungroup() %>%
  dplyr::select(FERTILIZER, everything(), -name)
Fit

## or
fun <- function(coefs = NA) {
  ## theta[1] is the precision
  return(Intercept + FERTILIZER * coefs[, "FERTILIZER"] +
    rnorm(nrow(coefs), sd = sqrt(1 / theta[1])))
}
Fit <- inla.posterior.sample.eval(fun, draws, coefs = newdata) %>%
  as.data.frame() %>%
  bind_cols(newdata) %>%
  pivot_longer(cols = -FERTILIZER) %>%
  group_by(FERTILIZER) %>%
  median_hdci(value)

Fit

Fitted (expected) values

fert.pred <- fert %>%
  bind_rows(newdata)
i.newdata <- (nrow(fert) + 1):nrow(fert.pred)
fert.inla2 <- inla(YIELD ~ FERTILIZER,
  data = fert.pred,
  family = "gaussian",
  control.fixed = list(
    mean.intercept = 80,
    prec.intercept = 0.00001,
    mean = 0,
    prec = 0.0384
  ),
  control.family = list(hyper = list(prec = list(
    prior = "loggamma",
    param = c(0.5, 0.31)
  ))),
  control.compute = list(config = TRUE, dic = TRUE, waic = TRUE, cpo = TRUE)
)

fert.inla2$summary.fitted.values[i.newdata, ]

## or on the link scale...
fert.inla2$summary.linear.predictor[i.newdata, ]

Linear combinations

Xmat <- model.matrix(~FERTILIZER, data = newdata)
lincomb <- inla.make.lincombs(as.data.frame(Xmat))

fert.inla3 <- inla(YIELD ~ FERTILIZER,
  data = fert,
  family = "gaussian",
  lincomb = lincomb,
  control.fixed = list(
    mean.intercept = 80,
    prec.intercept = 0.00001,
    mean = 0,
    prec = 0.0384
  ),
  control.family = list(hyper = list(prec = list(
    prior = "loggamma",
    param = c(0.5, 0.31)
  ))),
  control.compute = list(config = TRUE, dic = TRUE, waic = TRUE, cpo = TRUE)
)

fert.inla3$summary.lincomb.derived

Hypothesis testing

Since we have the entire posterior, we are able to make probability statements. We simply count up the number of MCMC sample draws that satisfy a condition (e.g represent a slope greater than 0) and then divide by the total number of MCMC samples.

For this exercise, we will explore the following:

• what this the probability that Yield increases with increasing Fertilizer concentration - this is just asking what proportion of the drawn slopes are greater than 0.
• how much of a change in yield do we expect if fertilizer concentration is increased from 100 to 200 units and what is the probability that the change is more than 50%

rstanarm

Prob. of effect

fert.rstanarm3 %>% hypothesis("FERTILIZER>0")

## Hypothesis Tests for class :
##         Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob
## 1 (FERTILIZER) > 0     0.81      0.09     0.66     0.94        Inf         1
##   Star
## 1    *
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.

Conclusions:

• the probability of a positive effect of Fertilizer on grass Yield is 1. That is, there is, we are 100% confident that there is an effect.
• the evidence ratio is normally the ratio of the number of cases that satisfy the hypothesis to the number of cases that do not. Since the number of cases that do not satisfy the hypothesis is 0, the evidence ratio is Inf (since division by 0)

Alternatively…

fert.rstanarm3 %>%
  tidy_draws() %>%
  summarise(P = sum(FERTILIZER > 0) / n())

## # A tibble: 1
## #   × 1
##       P
##   <dbl>
## 1     1

Conclusions:

• the probability of a positive effect of Fertilizer on grass Yield is 1. That is, there is, we are 100% confident that there is an effect.

Change 100 to 200 (Method 1)

newdata <- list(FERTILIZER = c(200, 100))
fert.rstanarm3 %>%
  emmeans(~FERTILIZER, at = newdata) %>%
  pairs()

##  contrast  estimate lower.HPD upper.HPD
##  200 - 100     80.9      63.4        98
## 
## Point estimate displayed: median 
## HPD interval probability: 0.95

Conclusions:

• the change in Yield associated with an increase in Fertilizer concentration from 100 to 200 is 80.93.

If we want to derive other properties, such as the percentage change, then we use tidy_draws() and then simple tidyverse spreadsheet operation.

fert.mcmc <- fert.rstanarm3 %>%
  emmeans(~FERTILIZER, at = newdata) %>%
  tidy_draws() %>%
  rename_with(~ str_replace(., "FERTILIZER ", "p")) %>%
  mutate(
    Eff = p200 - p100,
    PEff = 100 * Eff / p100
  )
fert.mcmc %>% head()

Now we can calculate the medians and HPD intervals of each column (and ignore the .chain, .iteration and .draw).

fert.mcmc %>% tidyMCMC(
  estimate.method = "median",
  conf.int = TRUE, conf.method = "HPDinterval"
)

Alternatively, we could use median_hdci

fert.mcmc %>% median_hdci(PEff)

Conclusions:

• the percentage change in Yield associated with an increase in Fertilizer concentration from 100 to 200 is 60.71% with a 95% credibility interval of 43.34 77.64

To get the probability that the effect is greater than a 50% increase.

fert.mcmc %>% summarise(P = sum(PEff > 50) / n())

## # A tibble: 1
## #   × 1
##       P
##   <dbl>
## 1 0.902

Conclusions:

• the probability that Yield will increase by more than 50 following an increase in Fertilizer concentration from 100 to 200 is 0.9.

Finally, we could alternatively use hypothesis(). Note that when we do so, the estimate is the difference between the effect and the hypothesised effect (50%).

fert.mcmc %>% hypothesis("PEff>50")

## Hypothesis Tests for class :
##        Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob
## 1 (PEff)-(50) > 0    10.83       8.8       -3    25.14       9.21       0.9
##   Star
## 1     
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.

### Change 100 to 200 (Method 2)


```r
newdata <- list(FERTILIZER = c(200, 100))
fert.rstanarm3 %>%
  emmeans(~FERTILIZER, at = newdata) %>%
  pairs() %>%
  tidy_draws() %>%
  summarise(across(contains("contrast"),
    list(
      P = ~ sum(. > 80) / n(),
      HDCI = ~ median_hdci(.)
    ),
    .names = c("{.fn}")
  )) %>%
  tidyr::unpack(HDCI)
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["P"],"name":[1],"type":["dbl"],"align":["right"]},{"label":["y"],"name":[2],"type":["dbl"],"align":["right"]},{"label":["ymin"],"name":[3],"type":["dbl"],"align":["right"]},{"label":["ymax"],"name":[4],"type":["dbl"],"align":["right"]},{"label":[".width"],"name":[5],"type":["dbl"],"align":["right"]},{"label":[".point"],"name":[6],"type":["chr"],"align":["left"]},{"label":[".interval"],"name":[7],"type":["chr"],"align":["left"]}],"data":[{"1":"0.54","2":"80.92855","3":"63.35079","4":"98.0094","5":"0.95","6":"median","7":"hdci"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>

We can also express this as a percentage change

```r
newdata <- list(FERTILIZER = c(200, 100))
## Simple
fert.rstanarm3 %>%
  emmeans(~FERTILIZER, at = newdata) %>%
  regrid(transform = "log") %>%
  pairs() %>%
  regrid()
```

```
##  contrast ratio lower.HPD upper.HPD
##  200/100   1.61      1.43      1.78
## 
## Point estimate displayed: median 
## HPD interval probability: 0.95
```

```r
## More advanced (both P and percent change)
fert.mcmc <- fert.rstanarm3 %>%
  emmeans(~FERTILIZER, at = newdata) %>%
  regrid(transform = "log") %>%
  pairs() %>%
  regrid() %>%
  tidy_draws() %>%
  mutate(across(contains("contrast"), ~ 100 * (. - 1)))

fert.mcmc %>%
  summarise(across(contains("contrast"),
    list(
      P = ~ sum(. > 50) / n(),
      HDCI = ~ median_hdci(.)
    ),
    .names = c("{.fn}")
  )) %>%
  tidyr::unpack(HDCI)
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["P"],"name":[1],"type":["dbl"],"align":["right"]},{"label":["y"],"name":[2],"type":["dbl"],"align":["right"]},{"label":["ymin"],"name":[3],"type":["dbl"],"align":["right"]},{"label":["ymax"],"name":[4],"type":["dbl"],"align":["right"]},{"label":[".width"],"name":[5],"type":["dbl"],"align":["right"]},{"label":[".point"],"name":[6],"type":["chr"],"align":["left"]},{"label":[".interval"],"name":[7],"type":["chr"],"align":["left"]}],"data":[{"1":"0.9020833","2":"60.7113","3":"43.34019","4":"77.64309","5":"0.95","6":"median","7":"hdci"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>

Change 100 to 200 (Method 3)

```r
fert.rstanarm3 %>%
  linpred_draws(newdata = as.data.frame(newdata)) %>%
  ungroup() %>%
  group_by(.draw) %>%
  summarise(
    Eff = -1 * diff(.linpred),
    PEff = 100 * Eff / .linpred[2]
  ) %>%
  ungroup() %>%
  mutate(P = sum(PEff > 50) / n()) %>%
  pivot_longer(cols = -.draw) %>%
  group_by(name) %>%
  median_hdci()
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["name"],"name":[1],"type":["chr"],"align":["left"]},{"label":["value"],"name":[2],"type":["dbl"],"align":["right"]},{"label":[".lower"],"name":[3],"type":["dbl"],"align":["right"]},{"label":[".upper"],"name":[4],"type":["dbl"],"align":["right"]},{"label":[".width"],"name":[5],"type":["dbl"],"align":["right"]},{"label":[".point"],"name":[6],"type":["chr"],"align":["left"]},{"label":[".interval"],"name":[7],"type":["chr"],"align":["left"]}],"data":[{"1":"Eff","2":"80.9285506","3":"63.3507924","4":"98.0094030","5":"0.95","6":"median","7":"hdci"},{"1":"P","2":"0.9020833","3":"0.9020833","4":"0.9020833","5":"0.95","6":"median","7":"hdci"},{"1":"PEff","2":"60.7113003","3":"43.3401878","4":"77.6430943","5":"0.95","6":"median","7":"hdci"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>

```r
## OR
fert.rstanarm3 %>%
  epred_draws(newdata = as.data.frame(newdata)) %>%
  ungroup() %>%
  group_by(.draw) %>%
  summarise(
    Eff = -1 * diff(.epred),
    PEff = 100 * Eff / .epred[2]
  ) %>%
  ungroup() %>%
  mutate(P = sum(PEff > 50) / n()) %>%
  pivot_longer(cols = -.draw) %>%
  group_by(name) %>%
  median_hdci()
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["name"],"name":[1],"type":["chr"],"align":["left"]},{"label":["value"],"name":[2],"type":["dbl"],"align":["right"]},{"label":[".lower"],"name":[3],"type":["dbl"],"align":["right"]},{"label":[".upper"],"name":[4],"type":["dbl"],"align":["right"]},{"label":[".width"],"name":[5],"type":["dbl"],"align":["right"]},{"label":[".point"],"name":[6],"type":["chr"],"align":["left"]},{"label":[".interval"],"name":[7],"type":["chr"],"align":["left"]}],"data":[{"1":"Eff","2":"80.9285506","3":"63.3507924","4":"98.0094030","5":"0.95","6":"median","7":"hdci"},{"1":"P","2":"0.9020833","3":"0.9020833","4":"0.9020833","5":"0.95","6":"median","7":"hdci"},{"1":"PEff","2":"60.7113003","3":"43.3401878","4":"77.6430943","5":"0.95","6":"median","7":"hdci"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>

```r
## OR for prediction of individual values
fert.rstanarm3 %>%
  predicted_draws(newdata = as.data.frame(newdata)) %>%
  ungroup() %>%
  group_by(.draw) %>%
  summarise(
    Eff = -1 * diff(.prediction),
    PEff = 100 * Eff / .prediction[2]
  ) %>%
  ungroup() %>%
  mutate(P = sum(PEff > 50) / n()) %>%
  pivot_longer(cols = -.draw) %>%
  group_by(name) %>%
  median_hdci()
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["name"],"name":[1],"type":["chr"],"align":["left"]},{"label":["value"],"name":[2],"type":["dbl"],"align":["right"]},{"label":[".lower"],"name":[3],"type":["dbl"],"align":["right"]},{"label":[".upper"],"name":[4],"type":["dbl"],"align":["right"]},{"label":[".width"],"name":[5],"type":["dbl"],"align":["right"]},{"label":[".point"],"name":[6],"type":["chr"],"align":["left"]},{"label":[".interval"],"name":[7],"type":["chr"],"align":["left"]}],"data":[{"1":"Eff","2":"80.05977","3":"15.451774","4":"136.04758","5":"0.95","6":"median","7":"hdci"},{"1":"P","2":"0.64625","3":"0.646250","4":"0.64625","5":"0.95","6":"median","7":"hdci"},{"1":"PEff","2":"60.21176","3":"8.853131","4":"133.64350","5":"0.95","6":"median","7":"hdci"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>

brms

Prob. of effect

fert.brm3 %>% hypothesis("FERTILIZER > 0")

## Hypothesis Tests for class b:
##         Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob
## 1 (FERTILIZER) > 0     0.81      0.06     0.72      0.9        Inf         1
##   Star
## 1    *
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.

fert.brm3 %>%
  hypothesis("FERTILIZER > 0") %>%
  plot(ignore_prior = TRUE)

fert.brm3 %>%
  hypothesis("FERTILIZER > 0") %>%
  `[[`("samples") %>%
  median_hdci()

Conclusions:

• the probability of a positive effect Fertilizer on grass yield is 1. That is, there is, we are 100% confident that there is an effect.
• the evidence ratio is normally the ratio of the number of cases that satisfy the hypothesis to the number of cases that do not. Since the number of cases that do not satisfy the hypothesis is 0, the evidence ratio is Inf (since division by 0)

Alternatively…

fert.brm3 %>%
  gather_draws(b_FERTILIZER) %>%
  summarise(P = sum(.value > 0) / n())

## # A tibble: 1
## #   × 2
##   .variable        P
##   <chr>        <dbl>
## 1 b_FERTILIZER     1

# OR
fert.brm3 %>%
  tidy_draws() %>%
  summarise(P = sum(b_FERTILIZER > 0) / n())

## # A tibble: 1
## #   × 1
##       P
##   <dbl>
## 1     1

Conclusions:

• the probability of a positive effect of Fertilizer on grass Yield is 1. That is, there is, we are 100% confident that there is an effect.

fert.brm3 %>% hypothesis("FERTILIZER>0.9")

## Hypothesis Tests for class b:
##               Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio
## 1 (FERTILIZER)-(0.9) > 0    -0.09      0.06    -0.18        0       0.05
##   Post.Prob Star
## 1      0.05     
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.

fert.brm3 %>%
  hypothesis("FERTILIZER>0.9") %>%
  plot(ignore_prior = TRUE)

# This returns a list
fert.brm3 %>%
  hypothesis("FERTILIZER>0.9") %>%
  plot(ignore_prior = TRUE) %>%
  `[[`(1) +
  geom_vline(xintercept = 0, linetype = "dashed")

# OR
fert.brm3 %>%
  tidy_draws() %>%
  summarise(P = sum(b_FERTILIZER > 0.9) / n())

## # A tibble: 1
## #   × 1
##        P
##    <dbl>
## 1 0.0492

fert.brm3 %>%
  tidy_draws() %>%
  ggplot(aes(x = b_FERTILIZER)) +
  geom_density(fill = "orange") +
  geom_vline(xintercept = 0.9, linetype = "dashed")

Change 100 to 200 (Method 1)

newdata <- list(FERTILIZER = c(200, 100))
fert.brm3 %>%
  emmeans(~FERTILIZER, at = newdata) %>%
  pairs()

##  contrast  estimate lower.HPD upper.HPD
##  200 - 100     80.8        69      91.2
## 
## Point estimate displayed: median 
## HPD interval probability: 0.95

Conclusions:

• the change in Yield associated with an increase in Fertilizer concentration from 100 to 200 is 80.82.

If we want to derive other properties, such as the percentage change, then we use tidy_draws() and then simple tidyverse spreadsheet operation.

It is in combination with emmeans() that tidy_draws() really comes into its own.

fert.mcmc <- fert.brm3 %>%
  emmeans(~FERTILIZER, at = newdata) %>%
  tidy_draws() %>%
  rename_with(~ str_replace(., "FERTILIZER ", "p")) %>%
  mutate(
    Eff = p200 - p100,
    PEff = 100 * Eff / p100
  )
fert.mcmc %>% head()

Now we can calculate the medians and HPD intervals of each column (and ignore the .chain, .iteration and .draw).

fert.mcmc %>%
  tidyMCMC(
    estimate.method = "median",
    conf.int = TRUE, conf.method = "HPDinterval"
  )

Alternatively, we could use median_hdci() to focus on a specific column.

fert.mcmc %>% median_hdci(PEff)

fert.mcmc %>%
  ggplot() +
  geom_density(aes(x = PEff))

Conclusions:

• the percentage change in Yield associated with an increase in Fertilizer concentration from 100 to 200 is 60.69% with a 95% credibility interval of 50.39 72.29

To get the probability that the effect is greater than a 50% increase.

fert.mcmc %>% summarise(P = sum(PEff > 50) / n())

## # A tibble: 1
## #   × 1
##       P
##   <dbl>
## 1 0.972

Conclusions:

• the probability that Yield will increase by more than 50 following an increase in Fertilizer concentration from 100 to 200 is 0.97.

Finally, we could alternatively use hypothesis(). Note that when we do so, the estimate is the difference between the effect and the hypothesised effect (50%).

fert.mcmc %>% hypothesis("PEff>50")

## Hypothesis Tests for class :
##        Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob
## 1 (PEff)-(50) > 0    10.79      5.53     1.79    20.01      34.29      0.97
##   Star
## 1    *
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.

Change 100 to 200 (Method 2)

newdata <- list(FERTILIZER = c(200, 100))
fert.brm3 %>%
  emmeans(~FERTILIZER, at = newdata) %>%
  pairs() %>%
  tidy_draws() %>%
  summarise(across(contains("contrast"),
    list(
      P = ~ sum(. > 80) / n(),
      HDCI = ~ median_hdci(.)
    ),
    .names = c("{.fn}")
  )) %>%
  tidyr::unpack(HDCI)

We can also express this as a percentage change

newdata <- list(FERTILIZER = c(200, 100))
## Simple
fert.brm3 %>%
  emmeans(~FERTILIZER, at = newdata) %>%
  regrid(transform = "log") %>%
  pairs() %>%
  regrid()

##  contrast ratio lower.HPD upper.HPD
##  200/100   1.61       1.5      1.72
## 
## Point estimate displayed: median 
## HPD interval probability: 0.95

## More advanced (both P and percent change)
fert.mcmc <- fert.brm3 %>%
  emmeans(~FERTILIZER, at = newdata) %>%
  regrid(transform = "log") %>%
  pairs() %>%
  regrid() %>%
  tidy_draws() %>%
  mutate(across(contains("contrast"), ~ 100 * (. - 1)))

fert.mcmc %>%
  summarise(across(contains("contrast"),
    list(
      P = ~ sum(. > 50) / n(),
      HDCI = ~ median_hdci(.)
    ),
    .names = c("{.fn}")
  )) %>%
  tidyr::unpack(HDCI)

Change 100 to 200 (Method 3)

fert.brm3 %>%
  linpred_draws(newdata = as.data.frame(newdata)) %>%
  ungroup() %>%
  group_by(.draw) %>%
  summarise(
    Eff = -1 * diff(.linpred),
    PEff = 100 * Eff / .linpred[2]
  ) %>%
  ungroup() %>%
  mutate(P = sum(PEff > 50) / n()) %>%
  pivot_longer(cols = -.draw) %>%
  group_by(name) %>%
  median_hdci()

## OR
fert.brm3 %>%
  epred_draws(newdata = as.data.frame(newdata)) %>%
  ungroup() %>%
  group_by(.draw) %>%
  summarise(
    Eff = -1 * diff(.epred),
    PEff = 100 * Eff / .epred[2]
  ) %>%
  ungroup() %>%
  mutate(P = sum(PEff > 50) / n()) %>%
  pivot_longer(cols = -.draw) %>%
  group_by(name) %>%
  median_hdci()

## OR for prediction of individual values
fert.brm3 %>%
  predicted_draws(newdata = as.data.frame(newdata)) %>%
  ungroup() %>%
  group_by(.draw) %>%
  summarise(
    Eff = -1 * diff(.prediction),
    PEff = 100 * Eff / .prediction[2]
  ) %>%
  ungroup() %>%
  mutate(P = sum(PEff > 50) / n()) %>%
  pivot_longer(cols = -.draw) %>%
  group_by(name) %>%
  median_hdci()

Summary figures

rstanarm

Simple

fert.list <- with(fert, list(FERTILIZER = seq(min(FERTILIZER), max(FERTILIZER), len = 100)))
newdata <- emmeans(fert.rstanarm3, ~FERTILIZER, at = fert.list) %>% as.data.frame()
head(newdata)

ggplot(newdata, aes(y = emmean, x = FERTILIZER)) +
  geom_point(data = fert, aes(y = YIELD)) +
  geom_line() +
  geom_ribbon(aes(ymin = lower.HPD, ymax = upper.HPD), fill = "blue", alpha = 0.3) +
  scale_y_continuous("YIELD") +
  scale_x_continuous("FERTILIZER") +
  theme_classic()

## spaghetti plot
newdata <- emmeans(fert.rstanarm3, ~FERTILIZER, at = fert.list) %>%
  gather_emmeans_draws()
newdata %>% head()

ggplot(newdata, aes(y = .value, x = FERTILIZER)) +
  geom_line(aes(group = .draw), alpha = 0.01) +
  geom_point(data = fert, aes(y = YIELD))

Pretty

fert.list <- with(fert, list(FERTILIZER = seq(min(FERTILIZER), max(FERTILIZER), len = 100)))
newdata <- emmeans(fert.rstanarm3, ~FERTILIZER, at = fert.list) %>% as.data.frame()
head(newdata)

ggplot(newdata, aes(y = emmean, x = FERTILIZER)) +
  geom_point(data = fert, aes(y = YIELD)) +
  geom_line() +
  geom_ribbon(aes(ymin = lower.HPD, ymax = upper.HPD), fill = "blue", alpha = 0.3) +
  scale_y_continuous(expression(Grass ~ yield ~ (g.m^-2)), breaks = seq(50, 300, by = 50)) +
  scale_x_continuous(expression(Fertilizer ~ concentration ~ (g.m^-2))) +
  theme_classic()

## spaghetti plot
newdata <- emmeans(fert.rstanarm3, ~FERTILIZER, at = fert.list) %>%
  gather_emmeans_draws()
newdata %>% head()

ggplot(newdata, aes(y = .value, x = FERTILIZER)) +
  geom_line(aes(group = .draw), colour = "orange", alpha = 0.01) +
  geom_point(data = fert, aes(y = YIELD)) +
  scale_y_continuous(expression(Grass ~ yield ~ (g.m^-2)), breaks = seq(50, 300, by = 50)) +
  scale_x_continuous(expression(Fertilizer ~ concentration ~ (g.m^-2))) +
  theme_classic()

brms

Simple

fert.grid <- with(fert, list(FERTILIZER = modelr::seq_range(FERTILIZER, n = 100)))
newdata <- fert.brm3 %>%
  emmeans(~FERTILIZER, at = fert.grid) %>%
  as.data.frame()
head(newdata)

ggplot(newdata, aes(y = emmean, x = FERTILIZER)) +
  geom_point(data = fert, aes(y = YIELD)) +
  geom_line() +
  geom_ribbon(aes(ymin = lower.HPD, ymax = upper.HPD), fill = "blue", alpha = 0.3) +
  scale_y_continuous("YIELD") +
  scale_x_continuous("FERTILIZER") +
  theme_classic()

## spaghetti plot
newdata <- emmeans(fert.brm3, ~FERTILIZER, at = fert.grid) %>%
  gather_emmeans_draws()
newdata %>% head()

ggplot(newdata, aes(y = .value, x = FERTILIZER)) +
  geom_line(aes(group = .draw), color = "blue", alpha = 0.01) +
  geom_point(data = fert, aes(y = YIELD))

Pretty

fert.list <- with(fert, list(FERTILIZER = seq(min(FERTILIZER), max(FERTILIZER), len = 100)))
newdata <- emmeans(fert.brm3, ~FERTILIZER, at = fert.list) %>% as.data.frame()
head(newdata)

ggplot(newdata, aes(y = emmean, x = FERTILIZER)) +
  geom_point(data = fert, aes(y = YIELD)) +
  geom_line() +
  geom_ribbon(aes(ymin = lower.HPD, ymax = upper.HPD), fill = "blue", alpha = 0.3) +
  scale_y_continuous(expression(Grass ~ yield ~ (g.m^-2)), breaks = seq(50, 300, by = 50)) +
  scale_x_continuous(expression(Fertilizer ~ concentration ~ (g.m^-2))) +
  theme_classic()

## spaghetti plot
newdata <- emmeans(fert.brm3, ~FERTILIZER, at = fert.list) %>%
  gather_emmeans_draws()
newdata %>% head()

ggplot(newdata, aes(y = .value, x = FERTILIZER)) +
  geom_line(aes(group = .draw), colour = "orange", alpha = 0.01) +
  geom_point(data = fert, aes(y = YIELD)) +
  scale_y_continuous(expression(Grass ~ yield ~ (g.m^-2)), breaks = seq(50, 300, by = 50)) +
  scale_x_continuous(expression(Fertilizer ~ concentration ~ (g.m^-2))) +
  theme_classic()

References

Fowler, J., L. Cohen, and P. Jarvis. 1998. Practical Statistics for Field Biology. England: John Wiley & Sons.